EXISTENCE AND STABILITY OF MULTIDIMENSIONAL TRANSONIC FLOWS THROUGH AN INFINITE NOZZLE OF ARBITRARY

CROSS-SECTIONS

GUI-QIANG CHEN AND MIKHAIL FELDMAN

Abstract. We establish the existence and stability of multidimensional transonic flows with transonic shocks through an infinite nozzle of arbitrary cross-sections, including a slowly varying de Lavel nozzle. The transonic flow is governed by the inviscid steady potential flow equation with supersonic upstream flow at the entrance, uniform subsonic downstream flow at the infinite exit, and the slip boundary condition on the nozzle boundary. The multidimensional transonic nozzle problem is reformulated into a free boundary problem, for which the free boundary is a transonic shock dividing two phases of C^{1,α} flow in the infinite nozzle, and the equation is hyperbolic in the supersonic upstream phase and elliptic in the subsonic downstream phase. We further develop a nonlinear iteration approach and employ its advantages to deal with such a free boundary problem in the unbounded domain and to solve the multidimensional transonic nozzle problem in a direct fashion. Our results indicate that, for the transonic nozzle problem, there exists a transonic flow such that the flow is divided into a C^{1,α} subsonic flow up to the nozzle boundary in the unbounded downstream region from the supersonic upstream flow by a C^{1,α} multidimensional transonic shock that is orthogonal to the nozzle boundary at every intersection point, and the uniform velocity state at the infinite exit in the downstream direction is uniquely determined by the supersonic upstream flow at the entrance which is sufficiently close to a uniform flow. The uniform velocity state at the exit can not be apriori prescribed from the corresponding pressure for such a flow to exist. We further prove that the transonic flow with a transonic shock is unique and stable with respect to the nozzle boundary and the smooth supersonic upstream flow at the entrance.

1. Introduction

We are concerned with the existence and stability of multidimensional steady transonic flows with transonic shocks through general multidimensional infinite nozzles with arbitrary crosssections. Such problems naturally arise in many physical situations, especially in the de Laval nozzles which was first proposed by Swedish engineer Gustav de Lavel in 1887 and has widely been used in the design of steam turbines and modern rocket engines (see Courant-Friedrichs [12], Whitham [48], and the references cited therein). Since the nozzles in applications are usually much longer with respect to their cross-sections, and hence the problem is often formulated mathematically as an infinite nozzle problem. Correspondingly, such a multidimensional infinite nozzle problem has extensively been studied experimentally, computationally, and asymptotically (see [12, 16, 19, 27, 22, 23, 41, 42, 48] and the references cited therein). Mathematically, the existence and stability of steady transonic flows for such nozzles in a multidimensional setup has been opened since then; see [5, 12, 13, 44, 48].

In this paper, we focus on the infinity nozzle to establish the existence and stability of multidimensional transonic flows with supersonic upstream flows at the entrance, uniform subsonic

Date: March 2004. 1991 Mathematics Subject Classification. 35M10,35J65,35R35,76H05,76L05,35B45. Key words and phrases. Elliptic-hyperbolic, nonlinear equations, second order, mixed type, multidimensional

transonic shocks, free-boundary problems, unbounded domains, infinite nozzles, cross-sections, existence, uniqueness, stability, Euler equations, inviscid potential flow.

downstream flows at the infinite exit, and the slip boundary condition on the nozzle boundary. The potential flow equation for the velocity potential φ : Ω ⊂ R^{n} → R is a second-order nonlinear equation of mixed elliptic-hyperbolic type:

div (ρ(|Dφ|^{2})Dφ) = 0,  x ∈ Ω ⊂ R^{n},  (1.1)

where the density ρ(q^{2}) is

ρ(q^{2}) = (1 − θq^{2})^{\frac{1}{2θ}}  (1.2)

and θ = \frac{γ−1}{2} > 0 with the adiabatic exponent γ > 1.

The nonlinear equation (1.1) is elliptic at Dφ with |Dφ| = q if

ρ(q^{2}) + 2q^{2}ρ′(q^{2}) > 0  (1.3)

and hyperbolic if

ρ(q^{2}) + 2q^{2}ρ′(q^{2}) < 0.  (1.4)

For this infinite nozzle problem, we first seek a multidimensional transonic flow containing a multidimensional transonic shock dividing a subsonic downstream flow from the supersonic upstream flow, motivated by our previous work [4, 5]. In order to construct such a transonic flow, we first formulate this infinite nozzle problem into a free boundary problem: the multidimensional transonic shock is a free boundary which divides two phases of C^{1,α} flow in the infinite nozzle, and the equation is hyperbolic in the supersonic upstream phase and elliptic in the subsonic downstream phase. Since the existence of the supersonic upstream phase is a direct corollary of the standard local existence theory of the initial-boundary value problem with the slip boundary condition for nonlinear wave equations (cf. [28, 29, 31]), we can further formulate the free boundary problem into a one-phase free boundary problem for a nonlinear elliptic equation by shiffmanization, a cut-off technique (see [4, 46]).

We further develop a nonlinear iteration approach based on [4] and employ its advantages to deal with such a free boundary problem in the unbounded domain and to solve the multidimensional transonic nozzle problem in a direct fashion. Our results indicate that, for the transonic nozzle problem, there exists a transonic flow such that the flow is divided into a C^{1,α} subsonic flow up to the nozzle boundary in the unbounded downstream region from the supersonic upstream flow by a C^{1,α} multidimensional transonic shock that is orthogonal to the nozzle boundary at every intersection point, and the uniform velocity state at the infinite exit in the downstream direction is uniquely determined by the supersonic upstream flow at the entrance which is sufficiently close to a uniform flow. The uniform velocity state at the exit can not be apriori prescribed from the corresponding pressure for such a flow to exist. We further prove that the transonic flow with a transonic shock is unique and stable with respect to the nozzle boundary and the smooth supersonic upstream flow at the entrance.

One of the advantages to employ our nonlinear iteration approach is that, as long as we know how the corresponding fixed conormal problem for (1.1) can be solved and estimated, the solution of the free boundary problem and its estimates directly follow, even for more complicated geometry of the domain under consideration. In this sense, the iteration approach is more efficient than the partial hodograph approach developed in [6], since we do not need to change equation (1.1) with fine features. This indeed plays an important role to achieve the C^{1,α} regularity of both the solution and the free boundary up to the nozzle boundary and to allow the nozzle boundary to have arbitrary cross-sections, by employing the features of equation (1.1). Also, this approach enables us to deal with the multidimensional nozzle problem in a direct fashion, which especially applies to the bounded nozzle problems with various different boundary conditions at the exit by solving the elliptic problems with nonsmooth fixed boundaries, as discussed in Section 3 and in [10, 11, 14, 24].

We point out in passing an important technical difference between the situation considered in this paper and the one in our previous work [4, 5]. In [5], we considered a transonic flow in an infinite cylinder with a cross-section of the form Λ = (0, 1)^{n−1}, i.e., a flow in the domain (0, 1)^{n−1} ×R, in which, by reflections, we can reduce the problem to the domain T^{n−1} ×R, where T^{n−1} is the (n − 1)-dimensional torus. Thus, in studying the free boundary problems in [4, 5], we were able to avoid studying the intersection of the free boundary with the fixed boundary. In the case of arbitrary smooth cross-section in dimension n > 2, this can not be done even for the flow in the unperturbed cylinder: indeed, if we locally flatten the boundary, the equation in the new coordinates changes its structure and the reflection leads to an equation with discontinuous coefficients. When we consider a flow in the nozzle (i.e. in a perturbed cylinder), we face a similar problem even in the two-dimensional case. Thus, in the present paper, we have to consider the intersection of a free boundary with a fixed boundary, that is, a nonlinear conormal problem in the domain with nonsmooth boundary.

The uniqueness and stability of the nozzle problem is a question of great importance to know under what circumstances a steady transonic flow involving transonic shocks is uniquely determined by the boundary condition and the conditions at the entrance and when further conditions at the exit are appropriate (see Courant-Friedrich [12]). In this paper, we first identify that the steady transonic flow involving a transonic shock is uniquely determined by the Cauchy data at the entrance for the nonlinear wave equation and the uniform subsonic flow condition at the infinite exit, which are natural physical conditions. Note that the pressure can not be apriori prescribed at the exit; otherwise, it is clear from our results that there is no weak solution for this problem.

Some efforts have been made for solving the nonlinear equation (1.1) of mixed type. In particular, Shiffman [46], Bers [2], and Finn-Gilbarg [18] proved the existence and uniqueness of solutions for the problem of subsonic flows of (1.1) past an obstacle (also see [15]). Morawetz in [43] showed that the flows of (1.1) past the obstacle may contain transonic shocks in general. In [4, 5, 6], we introduced two approaches to deal with multidimensional transonic shocks. Nontransonic shocks (hyperbolic-hyperbolic shocks) were analyzed in [7, 25, 33, 40, 45, 49] and the references cited therein.

In Section 2, we set up the infinite nozzle problem involving multidimensional transonic shocks and present the main theorems of this paper. In Section 3, we develop a C^{1,α} estimate framework up to the wedge boundary of the cylinder for the conormal problem for second-order linear elliptic equations of divergence form with C^{α} coefficients for subsequent development. In Section 4, we reformulate the transonic nozzle problem into a free boundary problem by introducing a multidimensional transonic shock as a free boundary which divides upstream and downstream phases of C^{1,α} flow in the infinite nozzle. We introduce an iteration procedure to construct approximate solutions of the free boundary problem in Section 4 and make uniform estimates of the solutions on the bounded domains with wedge boundary in Section 6. In Section 7, we establish the existence of multidimensional transonic flows with transonic shocks via solving the corresponding free boundary problem in the infinite nozzle. In Section 8, we establish the uniqueness and stability of multidimensional transonic flows with transonic shocks in the infinite nozzle.

The main results in this paper were presented in detail at the International Symposium on Multidimensional Conservation Laws and Related Topics (Shanghai, China), December 19-23, 2003; the Oberwolfach Workshop on Hyperbolic Conservation Laws (Germany), April 4-10, 2004; and the Stanford Workshop on Conservation Laws and Kinetic Theory (USA), July 17-29, 2004.

2. Transonic Shocks, Nozzle Problem, and Main Theorems

In this section, we first set up the infinite nozzle problem involving multidimensional transonic shocks and present the main theorems of this paper.

A function φ ∈ C^{0,1}(Ω) is a weak solution of (1.1) in an unbounded domain Ω if (i) |Dφ(x)| ≤ 1/\sqrt{θ} a.e. (ii) For any ζ ∈ C_{0}^{∞}(Ω),

∫_{Ω} ρ(|Dφ|^{2})Dφ · Dζ dx = 0.  (2.1)

We are interested in weak solutions with shocks. Let Ω^{+} and Ω^{−} be open subsets of Ω such that

Ω^{+} ∩ Ω^{−} = ∅, \overline{Ω^{+}} ∪ \overline{Ω^{−}} = \overline{Ω},

and S = ∂Ω^{+} ∩ Ω. Let φ ∈ C^{0,1}(Ω) be a weak solution of (1.1) and be in C^{1}(\overline{Ω^{±}}) so that Dφ experiences a jump across S that is an (n − 1)-dimensional smooth surface. Then φ satisfies the following Rankine-Hugoniot conditions on S:

[φ]_{S} = 0,  [ρ(|Dφ|^{2})Dφ · ν]_{S} = 0,  (2.2)

where ν is the unit normal to S from Ω^{−} to Ω^{+}, and the bracket denotes the difference between the values of the function along S on the Ω^{±} sides. Moreover, a function φ ∈ C^{1}(\overline{Ω^{±}}), which satisfies |Dφ| ≤ 1/\sqrt{θ}, (2.2), and equation (1.1) in Ω^{±} respectively, is a weak solution of (1.1), i.e., φ satisfies (2.1) in the whole domain Ω.

Set φ^{±} = φ|_{Ω^{±}}. Then we can also write (2.2) as

φ^{+} = φ^{−}  on S  (2.3)

and

ρ(|Dφ^{+}|^{2})φ_{ν}^{+} = ρ(|Dφ^{−}|^{2})Dφ^{−} · ν on S, (2.4)

where φ_{ν}^{+} = Dφ^{+} · ν is the normal derivative on the Ω^{+} side.

Note that the function

Φ(p) := (1 − θp^{2})^{\frac{1}{2θ}} p  (2.5)

is continuous on [0, \sqrt{1/θ}] and satisfies

Φ(p) > 0 for p ∈ (0, \sqrt{1/θ}) , Φ(0) = Φ (\sqrt{1/θ}) = 0,  (2.6)
0 < Φ′(p) < 1 on (0, c_{∗}),  Φ′(p) < 0 on (c_{∗}, \sqrt{1/θ}) ,  (2.7)
Φ′′(p) < 0 on (0, c_{∗}],  (2.8)

where

c_{∗} = \sqrt{1/(θ + 1)} = \sqrt{2/(γ + 1)}  (2.9)

is the sonic speed, for which a flow is called supersonic if |Dφ| > c_{∗} and subsonic if |Dφ| < c_{∗}. Suppose that φ ∈ C^{1}(\overline{Ω^{±}}) is a weak solution satisfying

|Dφ| < c_{∗} in Ω^{+},  |Dφ| > c_{∗} in Ω^{−},  Dφ^{±} · ν|_{S} > 0.  (2.10)

Then φ is a transonic shock solution with transonic shock S dividing Ω into the subsonic region Ω^{+} and the supersonic region Ω^{−} and satisfying the physical entropy condition (see Courant-Friedrichs [12]):

ρ(|Dφ^{−}|^{2}) < ρ(|Dφ^{+}|^{2}) along S.

Note that equation (1.1) is elliptic in the subsonic region and hyperbolic in the supersonic region.

Let (x′, x_{n}) be the coordinates in R^{n}, where x_{n} ∈ R and x′ = (x_{1}, …, x_{n−1}) ∈ R^{n−1}. Let q_{0}^{−} ∈ (c_{∗}, 1/\sqrt{θ}) and φ_{0}^{−}(x) := q_{0}^{−}x_{n}. Then φ_{0}^{−} is a supersonic solution in Ω. According to

(2.6)–(2.7), there exists a unique q_{0}^{+} ∈ (0, c_{∗}) such that

ρ ((q_{0}^{+})^{2}) q_{0}^{+} = ρ ((q_{0}^{−})^{2}) q_{0}^{−}.  (2.11)

Thus, the function

φ_{0}(x) = \{ \begin{matrix}q_{0}^{−}x_{n}, & x ∈ Ω_{0}^{−} := Ω ∩ \{x : x_{n} < 0\}, \\ q_{0}^{+}x_{n}, & x ∈ Ω_{0}^{+} := Ω ∩ \{x : x_{n} > 0\} \end{matrix}  (2.12)

is a plane transonic shock solution in Ω, Ω_{0}^{+} and Ω_{0}^{−} are its subsonic and supersonic regions respectively, and S = {x_{n} = 0} is a transonic shock.

Defining φ_{0}^{+}(x) := q_{0}^{+}x_{n} in Ω, we have

φ_{0}(x) = min(φ_{0}^{+}(x), φ_{0}^{−}(x)) for x ∈ Ω.  (2.13)

In this paper, we focus on the following infinite nozzle Ω with arbitrary smooth cross-sections:

Ω = Ψ(Λ × (−∞, ∞)) ∩ {x_{n} ≥ −1},  (2.14)

where Λ ⊂ R^{n−1} is an open bounded connected set with a smooth boundary, and Ψ : R^{n} → R^{n} is a smooth map, which is close to the identity map. For simplicity, we assume that

∂Λ is in C^{[ \frac{n}{2} ]+3,α}  (2.15)

and

∥Ψ − Id∥_{[ \frac{n}{2} ]+3,α,R^{n}} ≤ σ  (2.16)

for some α ∈ (0, 1) and small σ > 0, where [s] is the integer part of s, Id : R^{n} → R^{n} is the identity map, ∂_{l}Ω := Ψ(∂Λ × (−∞, ∞)) ∩ {(x′, x_{n}) : x_{n} > −1}, and ∥u∥_{m,α,D} is the norm in the Hölder space C^{m,α}(\overline{D}) in the domain D. Such nozzles especially include the slowly varying de Laval nozzles [12, 48]. For concreteness, we also assume that there exists L > L_{0} (say, L_{0} > 10 without loss of generality) such that

Ψ(x) = x  for any x = (x′, x_{n}) with x_{n} > L,  (2.17)

that is, the nozzle slowly varies locally in a bounded domain as the de Laval nozzles. In the two-dimension case, the domain Ω defined above has the following simple form:

Ω = {(x_{1}, x_{2}) : x_{1} ≥ −1, f_{−}(x_{2}) < x_{1} < f_{+}(x_{2})},

where ∥f_{±} − d_{±}∥_{4,α,R} ≤ σ and f_{±} ≡ d_{±} on [L, ∞) for some constants d_{±} satisfying d_{+} > d_{−}. For the multidimensional case, the geometry of the nozzles is much richer.

Note that our setup implies that

∂Ω = \overline{∂_{o}Ω} ∪ ∂_{l}Ω

with

∂_{l}Ω := Ψ[∂Λ × (−∞, ∞)] ∩ {(x′, x_{n}) : x_{n} > −1},
∂_{o}Ω := Ψ(Λ × (−∞, ∞)) ∩ {(x′, x_{n}) : x_{n} = −1}.

Then our transonic nozzle problem can be formulated into the following form:

Problem (TN): Transonic Nozzle Problem. Given the supersonic upstream flow at the entrance ∂_{o}Ω:

φ = φ_{e}^{−}, φ_{x_{n}} = ψ_{e}^{−}  on ∂_{o}Ω,  (2.18)

the slip boundary condition on the nozzle boundary ∂_{l}Ω:

∂_{ν}φ = 0  on ∂_{l}Ω,  (2.19)

and the uniform subsonic flow condition at the infinite exit x_{n} = ∞:

∥φ(·) − ωx_{n}∥_{C^{1}(Ω∩{x_{n}>R})} → 0  as R → ∞, for some ω ∈ (0, c_{∗}),  (2.20)

find a multidimensional transonic flow φ of the problem (1.1) and (2.18)–(2.20) in Ω.

As we will see below, the supersonic upstream part φ^{−} of the solution can be constructed in Ω_{1} := Ω ∩ {x_{n} < 1} from the data on the nozzle entrance (2.18) by using the standard results on initially-boundary value problems for hyperbolic equations. Thus, assuming that the C^{1,α} supersonic solution φ^{−} is given in Ω_{1}, we can formulate the transonic nozzle problem (TN) as the following one-phase free boundary problem.

Problem (FB): Free Boundary Problem. Given a supersonic upstream flow φ^{−}, a weak solution of (1.1) in Ω_{1}, which is a C^{1,α} perturbation of φ_{0}^{−}:

∥φ^{−} − φ_{0}^{−}∥_{1,α,Ω_{1}} ≤ C_{0}σ  (2.21)

with σ > 0 small, for some constant C_{0}, and satisfies (2.19), find a multidimensional subsonic flow φ^{+} of (1.1) satisfying (2.19)–(2.20) and identify a free boundary x_{n} = f(x′) dividing the subsonic flow φ^{+} from the given supersonic flow φ^{−} so that

φ(x) = \{ \begin{matrix}φ^{+}(x), & x_{n} > f(x′), \\ φ^{−}(x), & x_{n} < f(x′) \end{matrix}  (2.22)

is a transonic shock solution. Note that, for a solution φ of Problem (FB), the subsonic region Ω^{+} and supersonic region Ω^{−} in (2.10) are of the form

Ω^{+}(φ) = {x_{n} > f(x′)},  Ω^{−}(φ) = {x_{n} < f(x′)}.  (2.23)

Our main theorem for the free boundary problem, Problem (FB), is the following.

Theorem 2.1 (Existence). There exist σ_{0} > 0, C, and Ĉ, depending only on n, α, γ, q_{0}^{+}, C_{0}, Λ, and L, such that, for every σ ∈ (0, σ_{0}), any map Ψ satisfying (2.16) and (2.17), and any supersonic upstream flow φ^{−} of (1.1) satisfying (2.18) and (2.19), there exists a solution φ ∈ C^{0,1}(Ω) ∩ C^{∞}(Ω^{+}) of Problem (FB) such that

∥Dφ − q_{0}^{+}e_{n}∥_{0,0,Ω^{+}} ≤ Ĉσ.  (2.24)

Moreover, the solution φ satisfies the following properties:

(i) The constant ω in (2.20) must be q^{+}:

ω = q^{+},  (2.25)

where q^{+} is the unique solution in the interval (0, c_{∗}) of the equation

ρ((q^{+})^{2})q^{+} = Q^{+}  (2.26)

with

Q^{+} = \frac{1}{|Λ|} ∫_{∂_{o}Ω} ρ(|Dφ^{−}|^{2})Dφ^{−} · ν dH^{n−1}.  (2.27)

Thus φ satisfies

∥φ − q^{+}x_{n}∥_{C^{1}(Ω∩{x_{n}>R})} → 0 as R → ∞,  (2.28)

and q^{+} satisfies

|q^{+} − q_{0}^{+}| ≤ Cσ;  (2.29)

(ii) The function f(x′) in (2.22) satisfies

∥f∥_{1,α,R^{n−1}} ≤ Cσ,  (2.30)

and the surface S = {(x′, f(x′)) : x′ ∈ R^{n−1}} ∩ Ω is orthogonal to ∂_{l}Ω at every intersection point;

(iii) Furthermore, φ ∈ C^{1,α}(\overline{Ω^{+}}) with

∥φ − q^{+}x_{n}∥_{1,α,Ω_{+}} ≤ Cσ.  (2.31)

If the supersonic upstream flow φ^{−} has a higher regularity, then we have the following uniqueness theorem.

Theorem 2.2 (Uniqueness). There exists a constant σ_{0} > 0 depending only on n, α, γ, C_{0}, Λ, L, and q_{0}^{+} such that, if σ < σ_{0} and the supersonic solution φ^{−} in Problem (FB) additionally satisfies

∥φ^{−} − φ_{0}^{−}∥_{2,α,Ω_{1}} ≤ C_{0}σ,  (2.32)

then the solution φ of Problem (FB) satisfying (2.24) is unique.

The standard local existence theory of smooth solutions for the initial-boundary value problem (2.18)–(2.19) for second-order quasilinear hyperbolic equations implies (see Appendix) that, as σ is sufficiently small in (2.16) and (2.21), there exists a supersonic solution φ^{−} of (1.1) in Ω_{1}, which is a C^{k+1} perturbation of φ_{0}^{−}: For any α ∈ (0, 1]:

∥φ^{−} − φ_{0}^{−}∥_{k,α,Ω_{1}} ≤ C_{0}σ,  k = 1, 2,  (2.33)

for some constant C_{0} > 0, and satisfies

∂_{ν}φ^{−} = 0  on ∂_{l}Ω_{1},  (2.34)

provided that (φ_{e}^{−}, ψ_{e}^{−}) on ∂_{o}Ω satisfying

∥φ_{e}^{−} − q_{0}^{−}x_{n}∥_{H^{s+k}} + ∥ψ_{e}^{−} − q_{0}^{−}∥_{H^{s+k−1}} ≤ σ,  k = 1, 2,  (2.35)

for some integer s > n/2 + 1 and the compatibility conditions up to the (s + k)^{th}–order, where the norm ∥ · ∥_{H^{s}} is the Sobolev norm with H^{s} = W^{s,2}. In particular, as a direct corollary of Theorem 2.1, Theorem 2.2, and Proposition A.1 in Appendix, we obtain the following existence and uniqueness result for the transonic nozzle problem, Problem (TN):

Theorem 2.3. Let q_{0}^{+} ∈ (0, c_{∗}) and q_{0}^{−} ∈ (c_{∗}, 1/\sqrt{θ}) satisfy (2.11), and let φ_{0} be the transonic

shock solution (2.12). Then there exist σ_{0} > 0, C, and Ĉ, depending only on n, α, γ, q_{0}^{+}, Λ, and L such that, for every σ ∈ (0, σ_{0}), any map Ψ satisfying (2.16) and (2.17), and any supersonic upstream flow (φ_{e}^{−}, ψ_{e}^{−}) on ∂_{o}Ω satisfying

∥φ_{e}^{−} − q_{0}^{−}x_{n}∥_{H^{s+2}} + ∥ψ_{e}^{−} − q_{0}^{−}∥_{H^{s+1}} ≤ σ  (2.36)

for s > n/2 + 1 and the compatibility conditions up to the (s + 2)^{th}–order, there exists a unique solution φ ∈ C^{0,1}(Ω) of Problem (TN) satisfying (2.23),

∥φ − φ_{0}^{−}∥_{2,α,Ω^{−}} ≤ Cσ,

and (2.24). Moreover, this solution satisfies φ ∈ C^{0,1}(Ω) ∩ C^{∞}(Ω^{+}) and properties (i)–(iii) of Theorem 2.1, where (2.27) is replaced by

Q^{+} = \frac{1}{|Λ|} ∫_{∂_{o}Ω} ρ(|D_{x′}φ_{e}^{−}|^{2} + (ψ_{e}^{−})^{2})ψ_{e}^{−} dH^{n−1}.  (2.37)

Remark 2.1. The smoothness conditions (2.15) and (2.16) are just for simplicity of presentation and can be relaxed: For example, in Theorems 2.1 and 2.2, we require only C^{2,α} in (2.15) and (2.16). The localization condition (2.17) is not essential to achieve Theorem 2.1; In general, it can be replaced by an appropriate decay condition.

Remark 2.2. When the initial data (φ_{e}^{−}, ψ_{e}^{−}) ≡ (−ψ_{e}^{−}, ψ_{e}^{−}) is constant and the nozzle Ω∩{−1 ≤ x_{n} ≤ −1 + ϵ} = Λ × [−1, −1 + ϵ] for some ϵ > 0 as a de Laval nozzle, then the compatibility conditions are automatically satisfied. In fact, in this case, φ^{−}(x) = ψ_{e}^{−}x_{n} is a solution near x_{n} = −1 in the nozzle.

Remark 2.3. When n = 2, condition (2.36) for the supersonic upstream flow (φ_{e}^{−}, ψ_{e}^{−}) on ∂_{o}Ω in Theorem 2.3 can be replaced by the C^{3}-condition:

∥φ_{e}^{−} − q_{0}^{−}x_{n}∥_{C^{3}} + ∥ψ_{e}^{−} − q_{0}^{−}∥_{C^{2}} ≤ σ.  (2.38)

This can be achieved by following arguments in Li-Yu [34].

Remark 2.4. There exist σ_{0} and C depending only on the data such that equation (2.26) with Q^{+} defined by (2.27) has a unique solution q^{+} ∈ (0, c_{∗}) satisfying (2.29).

This can be seen as follows: From (2.16), it follows that | |∂_{o}Ω| − |Λ| | ≤ Cσ and |ν − e_{n}| ≤ Cσ on ∂_{o}Ω, and hence that |ν · e_{n} − 1| ≤ Cσ on ∂_{o}Ω. Then

|Q^{+} − ρ((q_{0}^{+})^{2})q_{0}^{+}|

= \frac{1}{|Λ|} ∫_{∂_{o}Ω} |ρ(|Dφ^{−}|^{2})Dφ^{−} · ν − ρ((q_{0}^{+})^{2})q_{0}^{+}| dx′ + (1 − \frac{|∂_{o}Ω|}{|Λ|}) ρ((q_{0}^{+})^{2})q_{0}^{+}

= \frac{1}{|Λ|} ∫_{∂_{o}Ω} |ρ(|Dφ^{−}|^{2})Dφ^{−} · ν − ρ((q_{0}^{−})^{2})q_{0}^{−}| dx′ + (1 − \frac{|∂_{o}Ω|}{|Λ|}) ρ((q_{0}^{+})^{2})q_{0}^{+}

≤ \frac{1}{|Λ|} ∫_{∂_{o}Ω} (|ρ(|Dφ^{−}|^{2})Dφ^{−} · ν − ρ(|Dφ_{0}^{−}|^{2})Dφ_{0}^{−} · ν| + ρ((q_{0}^{−})^{2})q_{0}^{−}|ν · e_{n} − 1|) dx′ + Cσ

≤ Cσ,

where we used (2.11) and (2.21). Thus, by (2.6) and (2.7), we obtain that, if σ is small depending only on the data, then there exists a unique solution q^{+} ∈ (0, c_{∗}) of equation (2.26) such that (2.29) holds.

Remark 2.5. In fact, the solutions with transonic shocks of Problem (TN) in Theorem 2.3 are also stable with respect to the nozzle boundaries and the smooth supersonic upstream flows at the entrance; see Theorem 8.1.

Remark 2.6. For the isothermal gas γ = 1, the same results can be obtained by following the same arguments in this paper.

3. Conormal Problems for Linear Elliptic Equations in a Cylinder

In this section, we develop a C^{1,α} estimate framework up to the boundary of the cylinder for a conormal problem for linear elliptic equations of divergence form with C^{α} coefficients.

Let Λ ⊂ R^{n−1} be an open bounded, connected set with C^{2,α} boundary. For Q > 0, denote

C_{Q} := Λ × (0, Q), Σ_{Q}^{1} := Λ × {x_{n} = 0}, Σ_{Q}^{2} := ∂Λ × (0, Q), Σ_{Q}^{3} := Λ × {x_{n} = Q}.

Consider the problem

∑_{i,j=1}^{n}(a_{ij}u_{x_{j}})_{x_{i}} = f  in C_{Q},  (3.1)

∑_{i,j=1}^{n} a_{ij}u_{x_{j}}ν_{i} = g_{k}  on Σ_{Q}^{k}, k = 1, 2,  (3.2)
u = h  on Σ_{Q}^{3},  (3.3)

where ν is the inward unit normal to Σ_{Q}^{1} and Σ_{Q}^{2} of C_{Q}.

The main issue in the argument below is to show the existence and uniqueness of solutions which are C^{1,α} up to the edge ∂Λ × {0} of the cylinder. Lieberman [37] studied general oblique derivative problems for linear elliptic equations in the domains with wedges in which the C^{1,β} regularity of solutions is obtained near the wedges. However, these results require certain conditions on the relative geometry of the wedge and the vector fields in the oblique derivative conditions; Moreover, in [37], the Hölder exponent β of the solution gradients depends on the geometry of wedges and the coefficients of the equation and the right-hand sides of the boundary conditions, and β < α in general where α is the Hölder exponent of the coefficients and the right-hand sides. In this section, by restricting the class of equations and considering only conormal boundary conditions and cylindrical domains, we obtain the sharp C^{1,α} regularity of solutions up to ∂Λ × {0}, i.e., β = α. This is important in our approach developed in Sections 4–8.

Theorem 3.1. Let λ_{1}, …, λ_{n}, and λ are the constants satisfying 0 < λ ≤ λ_{i} ≤ λ^{−1} for i = 1, …, n. Then there exist κ > 0 and C > 0, depending only on n, Λ, Q, and λ, such that, when a_{ij}(x) satisfy

∥a_{ij} − λ_{i}δ_{j}^{i}∥_{0,α,C_{Q}} ≤ κ for i, j = 1, …, n, (3.4)

and

f ∈ L^{∞}(C_{Q}), h ∈ H^{1}(C_{Q}),  (3.5)
g_{1} ∈ C^{0,α}(\overline{Σ_{Q}^{1}}), g_{2} ∈ C^{0,α}(\overline{Σ_{Q}^{2}}),  (3.6)

there exists a unique weak solution u ∈ H^{1}(C_{Q}) of (3.1)–(3.3) in the sense that (3.3) holds as the trace and

∫_{C_{Q}}( ∑_{i,j=1}^{n} a_{ij}u_{x_{j}}w_{x_{i}} + fw)dx + ∑_{k=1}^{2} ∫_{Σ_{Q}^{k}} g_{k}w dH^{n−1} = 0  (3.7)

for any function w ∈ H^{1}(C_{Q}) with w = 0 on ∂C_{Q} ∩ {x_{n} = Q}. Moreover, the solution u is C^{1,α}(\overline{C_{Q/2}}) and satisfies

∥u∥_{1,α,C_{Q/2}} ≤ C(∥u∥_{L^{2}(C_{Q})} + ∥f∥_{0,0,C_{Q}} + ∥g_{1}∥_{0,α,Σ_{Q}^{1}} + ∥g_{2}∥_{0,α,Σ_{Q}^{2}}).  (3.8)

Theorem 3.1 is a direct corollary of Theorem 3.2 below, stated in more technical terms. In order to state Theorem 3.2, we introduce some notations and weighted Hölder norms.

Denote

C′_{Q} = \overline{C_{Q}} \ \{x_{n} = Q\}, (Σ_{Q}^{2})′ = \overline{Σ_{Q}^{2}} \ \{x_{n} = Q\}. (3.9)

We will use the following weighted Hölder semi-norms and norms in the cylinder C_{Q}, in which the weight is the distance to the portion of the boundary \overline{Σ_{Q}^{3}} = ∂C_{Q} \ C′_{Q}. Denote δ_{x} = dist(x, \overline{Σ_{Q}^{3}}) = Q − x_{n} for x = (x′, x_{n}) ∈ C_{Q} and δ_{x,y} = min(δ_{x}, δ_{y}) for x, y ∈ C_{Q}. For k ∈ R, α ∈ (0, 1), and m ∈ N (the set of nonnegative integers), define

[[u]]_{m;0;C′_{Q}}^{(k)} = ∑_{|β|=m} sup_{x∈C′_{Q}} (δ_{x}^{m+k}|D^{β}u(x)|) ,

[[u]]_{m;α;C′_{Q}}^{(k)} = ∑_{|β|=m} sup_{x,y∈C′_{Q},x≠y} (δ_{x,y}^{m+α+k} \frac{|D^{β}u(x) − D^{β}u(y)|}{|x − y|^{α}}) ,

|u|_{m;0;C′_{Q}}^{(k)} = ∑_{j=0}^{m}[[u]]_{j;0;C′_{Q}}^{(k)},  (3.10)

|u|_{m;α;C′_{Q}}^{(k)} = |u|_{m;0;C′_{Q}}^{(k)} + [[u]]_{m;α;C′_{Q}}^{(k)},

where D^{β} = ∂_{x_{1}}^{β_{1}} ⋯ ∂_{x_{n}}^{β_{n}}, β = (β_{1}, …, β_{n}) is a multi-index with β_{j} ≥ 0, β_{j} ∈ N, and |β| = β_{1} + ⋯ + β_{n}. The weighted Hölder norms |u|_{m;α;(Σ_{Q}^{2})′}^{(k)} on the boundary part Σ_{Q}^{2} are similarly defined.

Theorem 3.2. Let λ_{1}, …, λ_{n}, and λ be the constants satisfying 0 < λ ≤ λ_{i} ≤ lambda^{−1} for i = 1, …, n. Then there exist κ > 0 and C > 0, depending only on n, Λ, Q, and λ, such that, when a_{ij}(x) satisfy (3.4) and

f ∈ L^{2}(C_{Q}) ∩ L_{loc}^{∞}(C′_{Q}), h ∈ H^{1}(C_{Q}),  (3.11)
g_{1} ∈ L^{2}(Σ_{Q}^{1}) ∩ C^{0,α}(\overline{Σ_{Q}^{1}}), g_{2} ∈ L^{2}(Σ_{Q}^{2}) ∩ C^{0,α}((Σ_{Q}^{2})′)  (3.12)

with

|f|_{0,0,C′_{Q}}^{(2+n/2)}, |g_{2}|_{0,α,(Σ_{Q}^{2})′}^{(1+n/2)} < ∞,

there exists a unique weak solution u ∈ H^{1}(C_{Q}) of (3.1)–(3.3) in the sense that (3.3) holds as the trace and (3.7) holds for any w ∈ H^{1}(C_{Q}) with w = 0 on ∂C_{Q} ∩ {x_{n} = Q}. Moreover, the solution u is C^{1,α}(C′) and satisfies

|u|_{1,α,C′_{Q}}^{(n/2)} ≤ C(∥u∥_{L^{2}(C_{Q})} + |f|_{0,0,C′_{Q}}^{(2+n/2)} + ∥g_{1}∥_{0,α,Σ_{Q}^{1}} + |g_{2}|_{0,α,(Σ_{Q}^{2})′}^{(1+n/2)}).  (3.13)

The remaining part of this section is a proof of Theorem 3.2. To prove the existence and regularity part of the theorem, we first consider an auxiliary problem:

∑_{i=1}^{n} λ_{i}u_{x_{i}x_{i}} = f + ∑_{i=1}^{n} ψ_{x_{i}}^{i}  in C_{Q},  (3.14)

∑_{i=1}^{n} λ_{i}u_{x_{i}}ν_{i} ≡ λ_{n}u_{x_{n}} = g_{1}  on Σ_{Q}^{1},  (3.15)

∑_{i=1}^{n} λ_{i}u_{x_{i}}ν_{i} ≡ ∑_{i=1}^{n−1} λ_{i}u_{x_{i}}ν_{i} = g_{2} + ∑_{i=1}^{n−1} ψ^{i}ν_{i}  on Σ_{Q}^{2},  (3.16)
u = h  on Σ_{Q}^{3},  (3.17)

where we used that ν = e_{n} on Σ_{Q}^{1} in (3.15) and ν · e_{n} = 0 on Σ_{Q}^{2} in (3.16). Note that smooth solutions of (3.14)–(3.17) satisfy

∫_{C_{Q}}(∑_{i=1}^{n} λ_{i}u_{x_{i}}w_{x_{i}} − ∑_{i=1}^{n} ψ^{i}w_{x_{i}} + fw)dx

+ ∫_{Σ_{Q}^{1}}(g_{1} − ψ^{n}) wdx′ − ∫_{Σ_{Q}^{2}} g_{2} w dH^{n−1} = 0  (3.18)

for any function w ∈ H^{1}(C_{Q}) with w = 0 on C_{Q} ∩ {x_{n} = Q}.

Lemma 3.1. Let λ_{1}, …, λ_{n}, and λ be the constants satisfying 0 < λ ≤ λ_{i} ≤ λ^{−1} for i = 1, …, n. Let

\begin{matrix}f ∈ L^{2}(C_{Q}) ∩ L_{loc}^{∞}(C′_{Q}), ψ = (ψ^{1}, …, ψ^{n}) ∈ H^{1}(C_{Q}) ∩ C^{α}(C′_{Q}), h ∈ H^{1}(C_{Q}), \\ g_{1} ∈ L^{2}(Σ_{Q}^{1}) ∩ C^{0,α}(\overline{Σ_{Q}^{1}}), g_{2} ∈ C(C′_{Q}) ∩ L^{2}(Σ_{Q}^{2}) ∩ C^{0,α}(C′_{Q}) \end{matrix}  (3.19)

with

|f|_{0,0,C′_{Q}}^{(2+n/2)}, |ψ|_{0,α,C′_{Q}}^{(1+n/2)}, |g_{2}|_{0,α,C′_{Q}}^{(1+n/2)} < ∞.  (3.20)

Then there exists a unique weak solution u ∈ H^{1}(C_{Q}) of (3.14)–(3.17) in the sense that (3.17) holds as the trace and (3.18) holds for any w ∈ H^{1}(C_{Q}) with w = 0 on ∂C_{Q} ∩ {x_{n} = Q}. The solution u is C^{1,α}(C′) and satisfies

∥u∥_{H^{1}(C_{Q})} + |u|_{1,α,C′_{Q}}^{(n/2)}
≤ C(∥f∥_{L^{2}(C_{Q})} + |f|_{0,0,C′_{Q}}^{(2+n/2)} + ∥ψ∥_{L^{2}(C_{Q})} + |ψ|_{0,α,C′_{Q}}^{(1+n/2)} + ∥g_{1}∥_{0,α,Σ_{Q}^{1}}
+∥g_{2}∥_{L^{2}(Σ_{Q}^{2})} + |g_{2}|_{0,α,(Σ_{Q}^{2})′}^{(1+n/2)} + ∥h∥_{H^{1}(C_{Q})})  (3.21)

and

|u|_{1,α,C′_{Q}}^{(n/2)} ≤ C(∥u∥_{L^{2}(C_{Q})} + |f|_{0,0,C′_{Q}}^{(2+n/2)} + |ψ|_{0,α,C′_{Q}}^{(1+n/2)} + ∥g_{1}∥_{0,α,Σ_{Q}^{1}} + |g_{2}|_{0,α,(Σ_{Q}^{2})′}^{(1+n/2)}),  (3.22)

where C depends only on n, Λ, Q, and λ.

Proof. We prove this lemma in six steps. The constant C in the argument below is a universal constant depending only on n, Λ, Q, and λ, and may be different at each occurrence.

Step 1. First we show the uniqueness of weak solutions. Suppose u_{1} and u_{2} are two weak solutions of (3.14)–(3.17). Then u_{1} − u_{2} = 0 on Σ_{Q}^{3}. Thus, writing the weak forms (3.18) for u_{1} and u_{2} respectively and subtracting the expressions with w = u_{1} − u_{2} yield

0 = ∫_{C_{Q}} ∑_{i,j=1}^{n} λ_{i}|∂_{x_{i}}(u_{1} − u_{2})|^{2}dx.

Since u_{1} − u_{2} = 0 on Σ_{Q}^{3} and λ_{i} > 0, we conclude u_{1} = u_{2} a.e. in C_{Q}.

Step 2. We now focus on the existence of solutions and estimate (3.21). First we show that the problem can be reduced to the case where the right-hand side of the boundary condition (3.15) vanishes. We can extend g_{1} from Σ_{Q}^{1} = Λ × {0} to the whole hyperplane {(x′, 0) : x′ ∈ R^{n−1}} such that the extension g_{1}^{∗} ∈ C_{0}^{α}(R^{n−1}) satisfies ∥g_{1}^{∗}∥_{0,α,R^{n−1}} ≤ C∥g_{1}∥_{0,α,Λ}. Now, as in [21, page 124-125], we choose a nonnegative η ∈ C_{0}^{2}(R^{n−1}) with ∫_{R^{n−1}} η(y′)dy′ = 1 and define

G(x′, x_{n}) = λ_{n}^{−1}x_{n} ∫_{R^{n−1}} g_{1}^{∗}(x′ − x_{n}y′)η(y′)dy′.

Then G ∈ C^{1,α}(\overline{R_{+}^{n}}) with

∥G∥_{1,α,C_{Q}} ≤ C∥g_{1}^{∗}∥_{0,α,R^{n−1}} ≤ C∥g_{1}∥_{0,α,Λ}  (3.23)

and

λ_{n}G_{x_{n}} = g_{1}^{∗} = g_{1} on Λ × {x_{n} = 0}.

Thus, u is a weak solution of (3.14)–(3.17) if and only if v = u − G is a weak solution of the same problem with modified right-hand sides: f, ψ, g_{1}, and g_{2} are replaced by f̃ = f,ψ̃^{i} = ψ^{i} − λ_{i}G_{x_{i}}, g̃_{1} = 0, and g̃_{2} = g_{2}. Using (3.23), it is easy to check that (3.21) and (3.22) for v in terms of f̃, ψ̃, g̃_{1}, and g̃_{2} implies (3.21) and (3.22) for u in terms of f, ψ, g_{1}, and g_{2}.

Thus, from now on, we assume

g_{1} = 0 on Σ_{Q}^{1}.  (3.24)

Step 3. We also note that we can assume without loss of generality that

ψ^{n} = 0  on Σ_{Q}^{1}.  (3.25)

Indeed, if u is a weak solution of (3.14)–(3.17), then u is also a weak solution of (3.14)–(3.17) with ψ^{n} replaced by ψ̂^{n}: ψ̂^{n}(x′, x_{n}) = ψ^{n}(x′, x_{n}) − ψ^{n}(x′, 0) for (x′, x_{n}) ∈ C_{Q}. Since

∥ψ̂^{n}∥_{L^{2}(C_{Q})} + |ψ̂|_{0,α,C′_{Q}}^{(1+n/2)} ≤ C(∥ψ^{n}∥_{L^{2}(C_{Q})} + |ψ|_{0,α,C′_{Q}}^{(1+n/2)}),

it follows that (3.21) and (3.22) in terms of the right-hand sides with ψ^{n} replaced by ψ̂^{n} implies (3.21) and (3.22) in terms of the original functions.

Clearly, ψ̂^{n} satisfies (3.25). Thus we can assume (3.25) from now on.

Step 4. With (3.24) and (3.25), we extend the problem (3.14)–(3.17) to the cylinder C_{(−Q,Q)} = Λ × (−Q, Q). Set Σ_{(−Q,Q)}^{2} := ∂Λ × (−Q, Q).

Note that, with (3.24) and (3.25), u is a weak solution of (3.14)–(3.17) if (3.17) holds as the trace and, for any w ∈ H^{1}(C_{Q}) with w = 0 on ∂C_{Q} ∩ {x_{n} = Q},

∫_{C_{Q}}(∑_{i=1}^{n} λ_{i}u_{x_{i}}w_{x_{i}} − ∑_{i=1}^{n} ψ^{i}w_{x_{i}} + fw)dx − ∫_{Σ_{Q}^{2}} g_{2}w dH^{n−1} = 0.  (3.26)

We use the even reflection to extend u, f, h, g_{2}, ψ^{1}, …, ψ^{n−1} to C_{(−Q,Q)} and the odd reflection to extend ψ^{n} to C_{(−Q,Q)}. That is, for x′ ∈ Λ and x_{n} ∈ (0, Q), we define, by using the same notations for the original and extended functions,

\begin{matrix}φ(x′, −x_{n}) = φ(x′, x_{n})  for φ ∈ \{u, f, h, g_{2}, ψ^{1}, …, ψ^{n−1}\}, \\ ψ^{n}(x′, −x_{n}) = −ψ^{n}(x′, x_{n}). \end{matrix}  (3.27)

Obviously, for u ∈ H^{1}(C_{Q}) and f, ψ, h, and g_{2} as in Lemma 3.1 with (3.25) for ψ^{n}, the extended functions satisfy

\begin{matrix}u ∈ H^{1}(C_{(−Q,Q)}), f ∈ L^{2}(C_{(−Q,Q)}) ∩ L_{loc}^{∞}(C′_{(−Q,Q)}), ψ ∈ H^{1}(C_{(−Q,Q)}) ∩ C^{α}(C′_{(−Q,Q)}), \\ h ∈ H^{1}(C_{(−Q,Q)}), g_{2} ∈ C^{α}(Σ_{(−Q,Q)}^{2}) ∩ L^{2}(Σ_{(−Q,Q)}^{2}), \end{matrix}

where C′_{(−Q,Q)} = C_{(−Q,Q)} ∪ Σ_{(−Q,Q)}^{2}. Moreover, defining the weighted Hölder semi-norms and

norms | · |_{m;α;C′_{(−Q,Q)}}^{(k)} and | · |_{m;α;Σ_{(−Q,Q)}^{2}}^{(k)} by (3.10) with the supremums taken over the domains

C_{(−Q,Q)} (resp. Σ_{(−Q,Q)}^{2}) and

δ_{x} = dist(x, ∂C_{(−Q,Q)} \ Σ_{(−Q,Q)}^{2}) = min(Q − x_{n}, Q + x_{n}) for x = (x′, x_{n}) ∈ C_{(−Q,Q)},

we get

\begin{matrix}|f|_{0,0,C′_{(−Q,Q)}}^{(2+n/2)} ≤ C|f|_{0,0,C′_{Q}}^{(2+n/2)}, |ψ|_{0,α,C′_{(−Q,Q)}}^{(1+n/2)} ≤ C|ψ|_{0,α,C′_{Q}}^{(1+n/2)}, \\ |g_{2}|_{0,α,Σ_{(−Q,Q)}^{2}}^{(1+n/2)} ≤ C|g_{2}|_{0,α,C′_{Q}}^{(1+n/2)}, \end{matrix}  (3.28)

where the norms on the right-hand sides are finite by (3.20).

Then it follows from (3.26) and (3.27) that the extended functions satisfy

∫_{C_{(−Q,Q)}} (∑_{i=1}^{n} λ_{i}u_{x_{i}}w_{x_{i}} − ∑_{i=1}^{n} ψ^{i}w_{x_{i}} + fw) dx − ∫_{Σ_{(−Q,Q)}^{2}} g_{2}w dH^{n−1} = 0  (3.29)

for any w ∈ H^{1}(C_{(−Q,Q)}) satisfying w = 0 on {x_{n} = ±Q}. Indeed, (3.29) holds separately when the integration is over subdomains {x_{n} > 0} and {x_{n} < 0} (the last follows by the change of variable x_{n} → −x_{n}). Thus, the extended function u is a weak solution of

∑_{i=1}^{n} λ_{i}u_{x_{i}x_{i}} = f + ∑_{i=1}^{n} ψ_{x_{i}}^{i}  in C_{(−Q,Q)},  (3.30)

∑_{i=1}^{n} λ_{i}u_{x_{i}}ν_{i} ≡ ∑_{i=1}^{n−1} λ_{i}u_{x_{i}}ν_{i} = g_{2} + ∑_{i=1}^{n−1} ψ^{i}ν_{i}  on Σ_{(−Q,Q)}^{2},  (3.31)
u = h  on (Λ × {−Q}) ∪ (Λ × {Q}).  (3.32)

Conversely, if u is a weak solution of (3.30)–(3.32) (i.e. (3.29) holds for w as above), then u satisfies u(x′, −x_{n}) = u(x′, x_{n}) for x′ ∈ Λ and x_{n} ∈ (0, Q). To see this, we first note that a weak solution of (3.29) is unique: The proof similarly follows the proof of uniqueness for (3.1)–(3.3) by using the fact that, if u_{1} and u_{2} are two weak solutions of (3.30)–(3.32), then we can use w = u_{1} −u_{2} in (3.29). Now, the properties in (3.27) for f, h, ψ, and g_{2} imply that, if u(x′, x_{n}) is a solution of (3.29), then u(x′, −x_{n}) is also a solution. Thus, by uniqueness, u(x′, −x_{n}) = u(x′, x_{n}). Now, using the properties (3.27) for u, f, h, ψ, and g_{2}, we see that, for all the integrals in (3.29), the integrals over subdomains {x_{n} > 0} and {x_{n} < 0} are equal. Thus, (3.29) implies (3.26). Then, in order to solve (3.14)–(3.17), it suffices to solve (3.30)–(3.32) for the extended right-hand sides.

Step 5. We now derive some estimates for the weak solution u ∈ H^{1}(C_{(−Q,Q)}) of (3.30)–(3.32). We first derive the energy estimate. We can use w = u − h in (3.29) and then use the ellipticity and Hölder inequality to obtain

∥u∥_{H^{1}(C_{(−Q,Q)})} ≤ C(∥f∥_{L^{2}(C_{(−Q,Q)})} + ∥ψ∥_{L^{2}(C_{(−Q,Q)})}
+∥h∥_{H^{1}(C_{(−Q,Q)})} + ∥g_{2}∥_{L^{2}(Σ_{(−Q,Q)}^{2})}).  (3.33)

Note that, by the standard local regularity results for the conormal problem for the elliptic equations in the domain with C^{1,α} boundary (e.g. [36]), we get u ∈ C^{1,α}(C′_{(−Q,Q)}).

Now we derive the estimates for the Hölder norms of u. We first recall the standard estimates for the conormal problem for the elliptic equations in a unit ball B_{1} = B_{1}(0) and half-ball B_{1}^{+} = B_{1}(0) ∩ {x_{n} > 0}. If u ∈ H^{1}(B_{1}) is a weak solution of the equation

∑_{i,j=1}^{n}(a_{ij}u_{x_{j}})_{x_{i}} = f + ∑_{i=1}^{n} ψ_{x_{i}}^{i}  in B_{1},  (3.34)

where a_{ij} ∈ C^{α}(B_{1}) satisfy the ellipticity condition: there exists some λ > 0 such that, for any x ∈ B_{1},

λ|ξ|^{2} ≤ ∑_{i,j=1}^{n} a_{ij}(x)ξ_{i}ξ_{j} ≤ λ^{−1}|ξ|^{2}  for any ξ ∈ R^{n},  (3.35)

and, if f ∈ L^{∞}(B_{1}) and ψ ∈ C^{α}(\overline{B_{1}}), then u ∈ C^{1,α}(\overline{B_{1/2}}) with

∥u∥_{1,α,B_{1/2}} ≤ C(∥u∥_{L^{2}(B_{1})} + ∥f∥_{L^{∞}(B_{1})} + ∥ψ∥_{0,α,B_{1}}),  (3.36)

where C depends only on n, λ, and ∥a_{ij}∥_{0,α,B_{1}}. This follows from [21, Theorem 8.32], combined with [21, Theorem 8.17]. Similar estimates can be obtained for the conormal problem, in which u ∈ H^{1}(B_{1}^{+}) is a weak solution of (3.34) in B_{1}^{+} with the conormal condition

∑_{j=1}^{n} a_{nj}u_{x_{j}} = g + ψ^{n} on ∂′B_{1}^{+} := ∂B_{1}^{+} ∩ {x_{n} = 0},  (3.37)

where g ∈ C^{α}(\overline{B}^{+}). Then u ∈ C^{1,α}(\overline{B_{1/2}^{+}}) with

∥u∥_{1,α,B_{1/2}^{+}} ≤ C(∥u∥_{L^{2}(B_{1}^{+})} + ∥f∥_{L^{∞}(B_{1}^{+})} + ∥ψ∥_{0,α,B_{1}^{+}} + ∥g∥_{0,α,∂′B_{1}^{+}}),  (3.38)

where C depends on n, λ, and ∥a_{ij}∥_{0,α,B_{1}^{+}}. This can be inferred, e.g., by simplifying the estimates of [36] to the case of linear equations.

Let ρ ∈ (0, 1/2). By scaling, we obtain the corresponding local estimates for weak solutions of (3.34) in B_{ρ} and for weak solutions of the conormal problem (3.34) and (3.37) in B_{ρ}^{+}. We write only the estimate for the conormal problem in B_{ρ}^{+}, since the interior estimate in B_{ρ} differs only in that it does not have the terms with the norms of g on the right-hand side. Thus, if u is a weak solution of (3.34) in B_{ρ/2}^{+} and satisfies (3.37) in B_{ρ}^{+} ∩ \{x_{n} = 0\}, then u ∈ C^{1,α}(\overline{B_{ρ/2}^{+}}) with

\begin{matrix}∥u∥_{0,0,B_{ρ/2}^{+}} + ρ∥Du∥_{0,0,B_{ρ/2}^{+}} + ρ^{1+α}[Du]_{0,α,B_{ρ/2}^{+}} \\ ≤ C(ρ^{−n/2}∥u∥_{L^{2}(B_{ρ}^{+})} + ρ^{2}∥f∥_{L^{∞}(B_{ρ}^{+})} + ρ∥ψ∥_{0,0,B_{ρ}^{+}} \\ +ρ^{1+α}[ψ]_{0,α,B_{ρ}^{+}} + ρ∥g∥_{0,0,∂′B_{ρ}^{+}} + ρ^{1+α}[g]_{0,α,∂′B_{ρ}^{+}}). \end{matrix}  (3.39)

This estimate is obtained by introducing the function v(x) = u(ρx) in B_{1}^{+}, writing the equation and conormal condition satisfied by u in terms of v, using estimate (3.38) for v with the corresponding right-hand sides of the equation and conormal condition satisfied by v, and then rewriting that estimate in terms of u.

Now we get similar estimates for weak solutions of (3.30)–(3.32). Using that ∂Λ is C^{1,α} and compact, there exits r > 0 and M > 0 such that, for any z ∈ Σ_{(−Q,Q)}^{2} and any positive ρ < min(r, δ_{z}), there is a C^{1,α} diffeomorphism F that flattens Σ_{(−Q,Q)}^{2} in B_{ρ}(z) ∩ C_{(−Q,Q)}, i.e., B_{ρ/M}^{+} ⊂ F (B_{ρ}(z) ∩ C_{(−Q,Q)}) ⊂ B_{Mρ}^{+}, and ∥(F, F^{−1})∥_{C^{1,α}} depends only on Λ. The transformed function v(y) := u(F^{−1}(y)) satisfies a conormal problem of the form (3.34) and (3.37) in B_{ρ/M}^{+}, where the equation is elliptic (with ellipticity constant λ/2) and the C^{α} norms of the coefficients a_{ij} depending only on the C^{1,α} norms of ∂Λ, and the appropriate norms of right-hand sides of the original and transformed problems are estimated by one in terms of another with a constant depending only on the C^{1,α} norms of ∂Λ. This can be seen by choosing the function w supported in B_{ρ}(z) in (3.29) and changing variables x → y = F (x) in (3.29). Now we get estimate (3.39) for v, which implies the following estimate for u:

∥u∥_{0,0,Ω_{ρ/2}(z)} + ρ∥Du∥_{0,0,Ω_{ρ/2}(z)} + ρ^{1+α}[Du]_{0,α,Ω_{ρ/2}(z)}
≤ C(ρ^{−n/2}∥u∥_{L^{2}(Ω_{ρ}(z))} + ρ^{2}∥f∥_{L^{∞}(Ω_{ρ}(z))} + ρ∥ψ∥_{0,0,Ω_{ρ}(z)} + ρ^{1+α}[ψ]_{0,α,Ω_{ρ}(z)}
+ρ∥g_{2}∥_{0,0,∂′Ω_{ρ}(z)} + ρ^{1+α}[g_{2}]_{0,α,∂′Ω_{ρ}(z)})

(3.40)

for any z ∈ Σ_{(−Q,Q)}^{2}, 0 < ρ < min(r, δ_{z}), with C depending only on n, Λ, L, and λ, where Ω_{ρ}(z) := B_{ρ}(z) ∩ C_{(−Q,Q)} and ∂′Ω_{ρ}(z) := B_{ρ}(z) ∩ Σ_{(−Q,Q)}^{2}. We also have the corresponding interior estimates: for any z ∈ C_{(−Q,Q)} and 0 < ρ < dist(z, ∂C_{(−Q,Q)}), we obtain (3.40) for Ω_{ρ}(z) = B_{ρ}(z) without the terms involving the norms of g_{2} on the right-hand side. Multiplying the interior and boundary estimates (3.40) by ρ^{n/2} and using a standard argument (e.g., [21, Theorem 4.8]) yields

|u|_{1,α,C′_{(−Q,Q)}}^{(n/2)} ≤ C(∥u∥_{L^{2}(C_{(−Q,Q)})} + |f|_{0,0,C′_{(−Q,Q)}}^{(2+n/2)}  (3.41)
+|ψ|_{0,α,C′_{(−Q,Q)}}^{(1+n/2)} + |g_{2}|_{0,α,Σ_{(−Q,Q)}^{2}}^{(1+n/2)}).

Obviously, estimates (3.33) and (3.41) imply (3.21). Also, (3.41) implies (3.22).

Step 6. It remains to prove the existence of a weak solution u ∈ H^{1}(C_{(−Q,Q)}) of (3.30)–(3.32). We first assume that f, ψ ∈ C^{∞}(\overline{C_{(−Q,Q)}}) and g_{2} ∈ C^{∞}(\overline{Σ_{(−Q,Q)}^{2}}). Then, since Σ_{(−Q,Q)}^{2} is a C^{2,α}

surface and g_{2} + ∑_{i=1}^{n−1} ψ^{i}ν_{i} ∈ C^{1,α}(\overline{Σ_{(−Q,Q)}^{2}}), we can find G ∈ C^{2,α}(C′_{(−Q,Q)}) such that

∑_{i=1}^{n−1} λ_{i}G_{x_{i}}ν_{i} = g_{2} + ∑_{i=1}^{n−1} ψ^{i}ν_{i}  on Σ_{(−Q,Q)}^{2}.  (3.42)

To construct such a function G, we first extend the right-hand side of (3.42) to Σ_{(−2Q,2Q)}^{2} =

∂Λ × (−2Q, 2Q) so that g̃ ∈ C^{1,α}(\overline{Σ_{(−2Q,2Q)}^{2}}) with g̃ = g_{2} + ∑_{i=1}^{n−1} ψ^{i}ν_{i} on Σ_{(−Q,Q)}^{2}. Then, for every x_{∗} ∈ ∂Λ × (−3Q/2, 3Q/2), we can locally flatten Σ_{(−2Q,2Q)}^{2} by a C^{2,α} diffeomorphism Ψ : R^{n} → R^{n}, i.e., for some r > 0, Ψ(B_{r}(x_{∗}) ∩ (Λ × R)) = B_{1}(0) ∩ {x_{n} > 0} with Ψ(B_{r}(x_{∗}) ∩ (∂Λ × R)) = B_{1}(0) ∩ {x_{n} = 0}. Then condition (3.42) is transformed into

∑_{i=1}^{n} c_{i}(x′)Ĝ_{x_{i}} = ĝ  on {x_{n} = 0},  (3.43)

where c_{i} ∈ C^{1,α}(R^{n−1}) with c_{n} ≥ min(λ_{1}, …, λ_{n})/2 > 0 if r is chosen sufficiently small, and ĝ ∈ C^{1,α}(R^{n−1}) with compact support is obtained by transforming g̃ by Ψ and extending to R^{n−1}. Now, by the argument similar to [21, page 124-125], we choose a nonnegative η ∈ C_{0}^{2}(R^{n−1}) with ∫_{R^{n−1}} η(y′)dy′ = 1, define the function

Ĝ(x′, x_{n}) = x_{n} ∫_{R^{n−1}} \frac{ĝ(x′ − x_{n}y′)}{c_{n}(x′ − x_{n}y′)}η(y′)dy′,

and show that Ĝ is C^{2,α}(\overline{R_{+}^{n}}) and satisfies (3.43). Transforming Ĝ back by Ψ^{−1}, we obtain G ≡ G^{x_{∗}} ∈ C^{2,α}((\overline{Λ} × R) ∩ B_{r}(x_{∗})) satisfying (3.42) on Σ_{(−2Q,2Q)}^{2} ∩ B_{r}(x_{∗}). Gluing these local functions by using a partition of unity, we get a C^{2,α} function G in a neighborhood of the boundary part Σ_{(−Q,Q)}^{2} in C, which satisfies (3.42) on Σ_{(−Q,Q)}^{2}. Finally, we extend G to the whole cylinder C_{(−Q,Q)} with G ∈ C^{2,α}(C′_{(−Q,Q)}).

Now, denoting f̃ = f + ∑_{i=1}^{n}(ψ_{x_{i}}^{i} − λ_{i}G_{x_{i}x_{i}}), we have f̃ ∈ C(\overline{C_{(−Q,Q)}}). Thus we can solve the variational problem of minimizing

I[v] = ∫_{C_{(−Q,Q)}}(\frac{1}{2} ∑_{i=1}^{n} λ_{i}v_{x_{i}}^{2} + f̃v)dx

on the set {v ∈ H^{1}(C_{(−Q,Q)}) : v = h − G on Λ × {x_{n} = ±Q}}. If v is a minimizer, then u = v + G is a weak solution of (3.30)–(3.32).

For general f, ψ ∈ L^{2}(C_{(−Q,Q)}) and g_{2} ∈ L^{2}(Σ_{(−Q,Q)}^{2}), we approximate by f_{l}, ψ_{l} ∈ C^{∞}(\overline{C_{(−Q,Q)}})

and g_{2}^{l} ∈ C^{∞}(\overline{Σ_{(−Q,Q)}^{2}}) such that (f_{l}, ψ_{l}) → (f, ψ) in L^{2}(C_{(−Q,Q)}) and g_{2}^{l} → g_{2} in L^{2}(Σ_{(−Q,Q)}^{2}). Then, for each l, we can find a weak solution u_{l} ∈ H^{1}(C_{(−Q,Q)}) of (3.30)–(3.32) with functions f_{l}, ψ_{l}, and g_{2}^{l} on the right-hand side. Since u_{k} − u_{l} satisfies (3.30)–(3.32) with f_{k} − f_{l}, ψ_{k} − ψ_{l}, g_{2}^{k} − g_{2}^{l}, and h = 0 on the right-hand side, we apply (3.33) to see that {u_{l}} is a Cauchy sequence in H^{1}(C_{(−Q,Q)}) and its limit u is a solution of (3.30)–(3.32) with the original right-hand sides. Now Lemma 3.1 is proved. □

Proof of Theorem 3.2. If κ is small depending on λ, then (3.4) implies the ellipticity (3.35) with \frac{3}{4}λ instead of λ. We choose such κ below.

First we show the uniqueness of weak solutions. Suppose u_{1} and u_{2} are two weak solutions of (3.1)–(3.3). Then u_{1} − u_{2} = 0 on Σ_{Q}^{3}. Writing the weak form (3.7) for u_{1} and u_{2} respectively, subtracting the expressions with w = u_{1} − u_{2}, and then using the ellipticity yields

0 = ∫_{C_{Q}} ∑_{i,j=1}^{n} a_{ij}∂_{x_{i}}(u_{1} − u_{2})∂_{x_{j}}(u_{1} − u_{2})dx ≥ \frac{3}{4}λ ∫_{C_{Q}} |D(u_{1} − u_{2})|^{2}dx.

Since u_{1} − u_{2} = 0 on Σ_{Q}^{3}, we conclude u_{1} = u_{2} a.e. in C_{Q}.

We now apply the Banach contraction fixed point theorem to prove the existence and regularity of the weak solution u ∈ H^{1}(C_{Q}) of (3.1)–(3.3) for sufficiently small κ > 0 in the space:

K = {v ∈ H^{1}(C_{Q}) ∩ C^{1,α}(C′_{Q}) : ∥v∥_{K} := ∥v∥_{H^{1}(C_{Q})} + |v|_{1,α,C′_{Q}}^{(n/2)} < ∞}.

We now define a mapping J : K → K. For v ∈ K, consider the problem (3.14)–(3.17) with functions f̃, h̃, g̃_{1}, g̃_{2}, and ψ̃ on the right-hand sides, where

f̃ = f, h̃ = h, g̃_{2} = g_{2},  (3.44)
g̃_{1}(x) = g_{1}(x) + ∑_{i,j=1}^{n}(λ_{i}δ_{i}^{j} − a_{ij}(x))v_{x_{j}}(x)ν_{i}(x) for x ∈ Σ_{Q}^{1} = Λ × {0}, (3.45)

ψ̃^{i}(x) = ∑_{j=1}^{n}(λ_{i}δ_{i}^{j} − a_{ij}(x))v_{x_{j}}(x)  for x ∈ C_{Q}, i = 1, …, n,  (3.46)

where f, g_{1}, g_{2}, and h are from (3.1)–(3.3). Since ν = e_{n} on Σ_{Q}^{1}, we have

g̃_{1} = g_{1} + ∑_{j=1}^{n}(λ_{n}δ_{n}^{j} − a_{nj})v_{x_{j}} ≡ g_{1} + ψ̃^{n}  on x ∈ Σ_{Q}^{1}.  (3.47)

By (3.4) and v ∈ K, the functions g̃_{1} and ψ satisfy the conditions of Lemma 3.1. Thus, by Lemma 3.1, there exists a unique weak solution u of (3.14)–(3.17) with the right-hand sides described above which satisfies (3.21) and thus implies u ∈ K. We define the mapping J : K → K by setting Jv = u.

Now we show that J is a contraction mapping in the norm ∥ · ∥_{K} if κ > 0 is small. Let v^{1}, v^{2} ∈ K and u^{k} = Jv^{k} for k = 1, 2. Then u^{1} − u^{2} is a weak solution of (3.14)–(3.17) with the following functions f̂, ĥ, ĝ_{1}, ĝ_{2}, and ψ̂ on the right-hand sides:

f̂ = 0, ĝ_{2} = 0, ĥ = 0,

ψ̂^{i}(x) = ∑_{j=1}^{n}(λ_{i}δ_{i}^{j} − a_{ij}(x))(v_{x_{j}}^{1}(x) − v_{x_{j}}^{2}(x)) for x ∈ C_{Q}, i = 1, …, n,

ĝ_{1}(x) = ∑_{j=1}^{n}(λ_{n}δ_{n}^{j} − a_{nj}(x))(v_{x_{j}}^{1}(x) − v_{x_{j}}^{2}(x)) for x ∈ Σ_{Q}^{1}.

By (3.4), we get

∥ψ̂∥_{L^{2}(C_{Q})} ≤ Cκ∥Dv^{1} − Dv^{2}∥_{L^{2}(C_{Q})} ≤ Cκ∥v^{1} − v^{2}∥_{K};
|ψ̂|_{0,α,C′_{Q}}^{(1+n/2)} ≤ Cκ|Dv^{1} − Dv^{2}|_{0,α,C′_{Q}}^{(1+n/2)} ≤ Cκ|v^{1} − v^{2}|_{1,α,C′_{Q}}^{(n/2)} ≤ Cκ∥v^{1} − v^{2}∥_{K};
∥ĝ_{1}∥_{0,α,Σ_{Q}^{1}} ≤ Cκ∥Dv^{1} − Dv^{2}∥_{0,α,Σ_{Q}^{1}} ≤ Cκ|Dv^{1} − Dv^{2}|_{0,α,C′_{Q}}^{(1+n/2)}
≤ Cκ|v^{1} − v^{2}|_{1,α,C′_{Q}}^{(n/2)} ≤ Cκ∥v^{1} − v^{2}∥_{K}.

Thus, from (3.21),

∥u^{1} − u^{2}∥_{K} ≤ Cκ∥v^{1} − v^{2}∥_{K}.

Therefore, the mapping J : K → K is a contraction mapping in the norm ∥ · ∥_{K} if κ < 1/C, i.e., κ is small depending only on n, Λ, Q, and λ.

For such κ, there exists a fixed point u ∈ K satisfying Ju = u. Then u is a weak solution of (3.1)–(3.3). Indeed, since Ju = u, then u satisfies (3.18) with right-hand sides given by (3.44), (3.45), and (3.46) computed with v = u. Rearranging and taking into account the last expression in (3.47) yields (3.7). Also, since u satisfies (3.18) with right-hand sides given by (3.44), (3.45), and (3.46), u satisfies (3.22) with these right-hand sides, which implies, by estimating ψ̃ and g̃_{1} similar to the estimates of ψ̂ and ĝ_{1} above and with v = u:

|u|_{1,α,C′_{Q}}^{(n/2)} ≤ C(∥u∥_{L^{2}(C_{Q})} + |f|_{0,0,C′_{Q}}^{(2+n/2)} + ∥g_{1}∥_{0,α,Σ_{Q}^{1}} + |g_{2}|_{0,α,(Σ_{Q}^{2})′}^{(1+n/2)}) + Cκ|u|_{1,α,C′_{Q}}^{(n/2)}.

If κ is small, this implies (3.13). Theorem 3.2 is proved.

4. Free boundary problems in the infinite nozzle

In order to solve Problem (TN), we first reformulate it into a free boundary problem for the subsonic part of the solution, since Problem (TN) is originally hyperbolic-elliptic mixed.

We first modify equation (1.1) to make it uniformly elliptic so that it coincides with the original equation in the range Dφ in the subsonic region Ω^{+} for φ satisfying (2.31) with sufficiently small σ. The details of the truncation procedure are in [4, Section 4.2].

Let ε = (c_{∗} − q_{0}^{+})/2. Then there exists ρ̃ ∈ C^{1,1}([0, ∞)) and c_{j} > 0, j = 1, 2, 3, depending only on q_{0}^{+} and γ, such that

ρ̃(q^{2}) = ρ(q^{2})  if 0 ≤ q < c_{∗} − ε,  (4.1)
ρ̃(q^{2}) = c_{0} + \frac{c_{1}}{q}  if q > c_{∗} − ε,  (4.2)
0 < c_{0} ≤ (ρ̃(q^{2})q)′ ≤ c_{2}  for q ∈ (0, ∞).  (4.3)

Then the equation

L̃φ := div (ρ̃(|Dφ|^{2})Dφ) = 0  (4.4)

is uniformly elliptic, whose ellipticity constants depend only on q_{0}^{+} and γ.

To formulate the problem into a free boundary problem, the main point is to replace the pointwise gradient condition {|Dφ(x)| < c_{∗}} defining Ω^{+}(φ) by a condition stated in terms of φ so that our problem is formulated into the framework of free boundary problems.

To do that, we first need to construct a local C^{1,α} supersonic solution φ^{−} in the domain Ω_{1} := Ω ∩ {−1 < x_{n} < 1} of the initial-boundary value problem (2.18)–(2.19) for the nonlinear hyperbolic equation (1.1) such that

∥φ^{−} − φ_{0}^{−}∥_{1,α,Ω_{1}} ≤ C_{1}σ  (4.5)

when σ is sufficiently small. This is ensured by the local existence theorem, Proposition A.1 in Appendix. By the standard extension argument (see [5] and Section 7.2), we can extend φ^{−} to the whole infinite nozzle Ω such that

∥φ^{−} − φ_{0}^{−}∥_{1,α,Ω} ≤ C_{0}σ,  φ_{ν}^{−}|_{∂_{l}Ω} = 0.  (4.6)

Then the following heuristic observation motivates our formulation: By (2.13) and q_{0}^{−} > q_{0}^{+}, we have Ω^{+}(φ_{0}) = {x ∈ Ω : φ_{0}(x) < φ_{0}^{−}(x)}. Since φ^{−} is a small C^{1,α} perturbation of φ_{0}^{−} and q_{0}^{−} > q_{0}^{+}, then we expect that φ^{+} is close to φ_{0}^{+} in C^{1,α}(\overline{Ω^{+}(φ)}) so that we can expect that Ω^{+}(φ) = {x ∈ Ω : φ(x) < φ^{−}(x)}.

We also perform the corresponding truncation of the free boundary condition (2.4):

ρ̃(|Dφ|^{2})φ_{ν} = ρ(|Dφ^{−}|^{2})Dφ^{−} · ν  on S.  (4.7)

On the right-hand side of (4.7), we use the original function ρ since ρ ≠ ρ̃ on the range of |Dφ^{−}|^{2}. Note that (4.7), with the right-hand side considered as a known function, is the conormal boundary condition for the uniformly elliptic equation (4.4).

We first solve the following free boundary problem (TFB), which is a truncated version of Problem (FB), in which the gradient condition that determines Ω^{+} in Problem (TN) is also replaced by the condition φ ≤ φ^{−} in Ω with the definition Ω^{+} := {φ < φ^{−}} ∩ Ω.

Problem (TFB): Truncated Free Boundary Problem. Given a supersonic upstream flow φ^{−} ∈ C^{1,α}(Ω_{1}) of (1.1) satisfying (4.5), (2.18), and (2.19) for small σ > 0 and some constant C_{0}, find φ ∈ C(Ω) such that

(i) φ satisfies

φ ≤ φ^{−}  in Ω  (4.8)

and conditions (2.18) and (2.19) on the boundary and (2.20) at infinity; (ii) φ ∈ C^{1,α}(\overline{Ω^{+}}) ∩ C^{2}(Ω^{+}) is a solution of (4.4) in Ω^{+} := \{φ < φ^{−}\} ∩ Ω, the

non-coincidence set; (iii) the free boundary S = ∂Ω^{+} ∩ Ω is given by the equation x_{n} = f(x′) so that

Ω^{+} = {x_{n} > f(x′)} ∩ Ω with f ∈ C^{1,α}; (iv) the free boundary condition (4.7) holds on S.

In this paper we develop the iteration approach based on [4, 5], uniform estimate techniques at infinity, and the regularity estimate techniques near the nonsmooth boundary to construct a unique solution of Problem (TFB). Finally, we use an estimate for |Dφ| to conclude that the solution of the truncated problem, Problem (TFB), is actually a solution of Problem (FB) and thus Problem (TN).

5. Iteration Procedure and Uniform Estimates

We now introduce an iteration procedure to construct approximate solutions of Problem (TFB) in the domain Ω and make uniform estimates of the solutions on the unbounded domain with nonsmooth boundary.

5.1. Iteration Set. Let M ≥ 1. We set

K_{M} := \{ψ ∈ C^{1,α}(\overline{Ω}) : ∥ψ − q^{+}x_{n}∥_{1,α,Ω} ≤ Mσ\},  (5.1)

where q^{+} is defined by equation (2.26) (see Remark 2.4) and α ∈ (0, 1). From the definition, K_{M} is convex. Now we will show that K_{M} is compact in the weighted Hölder space C_{(−1)}^{1,α/2}(\overline{Ω}) as defined below.

For α ∈ (0, 1), m ∈ N (the set of nonnegative integers), and k ∈ R, the definition of the spaces C_{(k)}^{m,α}(\overline{Ω}) is

C_{(k)}^{m,α}(\overline{Ω}) := \{u ∈ C^{m,α}(\overline{Ω}) : ∥u∥_{m,α,Ω}^{(k)} < ∞\},  (5.2)

where ∥u∥_{m;α;Ω}^{(k)} = ∥u∥_{m;0;Ω}^{(k)} + [u]_{m;α;Ω}^{(k)} with

∥u∥_{m;0;Ω}^{(k)} = ∑_{0≤j≤m}( ∑_{|β|=j} sup_{x∈Ω} (δ_{x}^{k}|D^{β}u(x)|)),

[u]_{m;α;Ω}^{(k)} = ∑_{|β|=m} sup_{x,y∈Ω,x≠y} (δ_{x,y}^{k} \frac{|D^{β}u(x) − D^{β}u(y)|}{|x − y|^{α}}) ,

where δ_{x} = |x_{n}| + 1 for x = (x′, x_{n}) ∈ D and δ_{x,y} = min(δ_{x}, δ_{y}) for x, y ∈ D.

Lemma 5.1. If α > β > 0, then K_{M} is a compact subset of C_{(−1)}^{1,β}(\overline{Ω}).

Proof. Let ψ_{j} ∈ K_{M} for j = 1, 2, . . .. By a standard argument, we can extract a subsequence (still denoted by) ψ_{j}, which converges in C^{1,β} on every compact subset of \overline{Ω} to the limit ψ ∈ C^{1,α}(\overline{Ω}). Then ∥ψ − q^{+}x_{n}∥_{1,α,Ω} ≤ Mσ and thus ψ ∈ K_{M}. It remains to show that ∥ψ_{j} − ψ∥_{1,β,Ω}^{(−1)} → 0 as j → ∞.

Fix 0 < ε < 1. Then ∥ψ_{j}∥_{1,α,Ω∩{x_{n}>1/ε}}^{(−1)} ≤ (Mσ + q^{+})ε, and the same estimate holds for ψ. Also, since ψ_{j} → ψ in C^{1,β}(\overline{Ω} ∩ \{x_{n} ≤ 2/ε\}), there exists j_{0} such that, for j > j_{0}, ∥ψ_{j} − ψ∥_{1,β,Ω∩{x_{n}≤2/ε}}^{(−1)} ≤ ε. Then, for j > j_{0}, we have ∥ψ_{j} − ψ∥_{1,β,Ω}^{(−1)} ≤ Cε, and the assertion is proved. □

5.2. Construction of the Iteration Scheme. Let ψ ∈ K_{M}. Since q_{0}^{−} > q_{0}^{+}, it follows that, if

σ ≤ \frac{q_{0}^{−} − q_{0}^{+}}{C(M + 1)}  (5.3)

with large C depending only on n, then (4.5) implies

(φ^{−} − ψ)_{x_{n}}(x) ≥ \frac{q_{0}^{−} − q_{0}^{+}}{2} > 0 in Ω  (5.4)

and φ^{−} > ψ on {x_{n} > 1}. Then the set Ω^{+}(ψ) := {ψ < φ^{−}} ∩ Ω has the form:

Ω^{+}(ψ) = {x_{n} > f(x′)} ∩ Ω,  ∥f∥_{1,α,R^{n−1}} ≤ CMσ  (5.5)

with C depending only on q_{0}^{−} − q_{0}^{+}.

The inward unit normal to S_{ψ} := {x_{n} = f(x′)} ∩ Ω of Ω^{+} is

ν_{ψ}(x) = \frac{Dφ^{−}(x) − Dψ(x)}{|Dφ^{−}(x) − Dψ(x)|}  for x ∈ S_{ψ}.  (5.6)

By the definition of K_{M} and (5.3), formula (5.6) also defines ν_{ψ}(x) on Ω_{1} and

∥ν_{ψ} − e_{n}∥_{0,α,Ω_{1}} ≤ CMσ  with C = C(q_{0}^{+}, q_{0}^{−}).  (5.7)

Motivated by (4.7), we define the function

G_{ψ}(x) := ρ(|Dφ^{−}(x)|^{2})Dφ^{−}(x) · ν_{ψ}(x)  on Ω_{1}.  (5.8)

Then we consider the following elliptic problem in the domain Ω^{+}(ψ):

div (ρ̃(|Dφ|^{2})Dφ) = 0  in Ω^{+}(ψ),  (5.9)
ρ̃(|Dφ|^{2})φ_{ν} = G_{ψ}(x)  on S_{ψ} := {x_{n} = f(x′)},  (5.10)
φ_{ν} = 0  on ∂_{l}Ω^{+} := ∂Ω^{+}(ψ) ∩ ∂_{l}Ω,  (5.11)
lim_{R→∞} ∥φ − q^{+}x_{n}∥_{L^{∞}(Ω^{+}(ψ)∩{x_{n}>R})} = 0,  (5.12)

show that there exists a unique solution φ(x), and extend the solution to the whole domain Ω so that φ ∈ K_{M}.

We rewrite problem (5.9)–(5.12) in terms of function u(x) = φ(x) − q^{+}x_{n}. We first note that the boundary condition (5.11) is equivalent to

ρ̃(|Dφ|^{2})φ_{ν} = 0 on ∂_{l}Ω^{+}.

Now u(x) is a solution of the following problem:

div A(Du) = 0  in Ω^{+}(ψ),  (5.13)
A(Du) · ν = g_{ψ}(x)  on S_{ψ},  (5.14)
A(Du) · ν = −ρ̃((q^{+})^{2})q^{+}ν · e_{n}  on ∂_{l}Ω^{+},  (5.15)
lim_{R→∞} ∥u∥_{L^{∞}(Ω^{+}(ψ)∩{x_{n}>R})} = 0,  (5.16)

where

A(P) = ρ̃(|P + q^{+}e_{n}|^{2})(P + q^{+}e_{n}) − ρ̃((q^{+})^{2})q^{+}e_{n} for P ∈ R^{n},  (5.17)
g_{ψ}(x) = G_{ψ}(x) − ρ((q^{+})^{2})q^{+}ν · e_{n}.  (5.18)

Thus, u(x) satisfies the uniformly elliptic equation with the same ellipticity constants as in (4.4):

λ|ξ|^{2} ≤ ∑_{i,j=1}^{n} A_{P_{j}}^{i}(P)ξ_{i}ξ_{j} ≤ λ^{−1}|ξ|^{2}  for any P, ξ ∈ R^{n}.  (5.19)

Moreover, from the definition of A(P), (4.1), and (4.2), A(P) satisfies

A(0) = 0,  (1 + |P |)|D_{P} A_{P_{j}}^{i}(P)| ≤ C.  (5.20)

Finally, we state a linear problem corresponding to (5.13)–(5.16). Namely, we use (5.20) to find that, for i = 1, …, n,

A^{i}(Du(x)) = ∑_{j=1}^{n} ã_{ij}(x)u_{x_{j}}(x), ã_{ij}(x) = ∫_{0}^{1} A_{p_{j}}^{i}(tDu(x))dt.

We replace u ≡ φ − q^{+}x_{n} in the definition of the coefficients ã_{ij} by ψ − q^{+}x_{n} for ψ ∈ K_{M} to define

a_{ij}(x) = a_{ij}^{(ψ)}(x) = ∫_{0}^{1} A_{p_{j}}^{i} (t(Dψ(x) − q^{+}e_{n})) dt  for x ∈ Ω, i, j = 1, …, n. (5.21)

The ellipticity (5.19) of A(P) implies that the coefficients {a_{ij}^{(ψ)}(x)} satisfy the ellipticity condition (3.35) for any x ∈ Ω.

Also, from (5.17) and (5.21),

a_{ij}^{(ψ)}(x)  (5.22)
= ∫_{0}^{1} {ρ̃(|tDψ(x) + (1 − t)q^{+}e_{n}|^{2})δ_{i}^{j}
+2ρ̃′(|tDψ(x) + (1 − t)q^{+}e_{n}|^{2})(tψ_{x_{i}}(x) + (1 − t)q^{+}δ_{i}^{n})(tψ_{x_{j}}(x) + (1 − t)q^{+}δ_{j}^{n})}dt

for x ∈ Ω, where δ_{i}^{j} = 1 if i = j and δ_{i}^{j} = 0 if i ≠ j. In particular, we have a_{ij} = a_{ji}.

We note that, for ψ̆_{0} = q^{+}x_{n}, the corresponding coefficients ă_{ij} defined by (5.22) are constants and satisfy

ă_{ij} = κ_{i}δ_{i}^{j} for i, j = 1, …, n, (5.23)

where κ_{i} = ρ̃((q^{+})^{2}) for i = 1, …, n − 1; and κ_{n} = ϕ′(q^{+}) for ϕ(s) = ρ̃(s^{2})s. We have

λ ≤ κ_{i} ≤ λ^{−1} for i = 1, …, n.

Now, for ψ ∈ K_{M}, we use (5.22) and (4.1)–(4.3) to obtain

∥a_{ij}^{(ψ)} − ă_{ij}∥_{0,α,Ω} ≤ CMσ.  (5.24)

Thus we formulate the following conormal fixed boundary elliptic problem:

∑_{i,j=1}^{n}(a_{ij}^{(ψ)}u_{x_{j}})_{x_{i}} = 0  in Ω^{+}(ψ),  (5.25)

∑_{i,j=1}^{n} a_{ij}^{(ψ)}u_{x_{j}}ν_{i} = g_{ψ}(x)  on S_{ψ},  (5.26)

∑_{i,j=1}^{n} a_{ij}^{(ψ)}u_{x_{j}}ν_{i} = −ρ̃((q^{+})^{2})q^{+}ν · e_{n}  on ∂_{l}Ω^{+}(ψ),  (5.27)
lim_{R→∞} ∥u∥_{C(Ω^{+}(ψ)∩{x_{n}>R})} = 0.  (5.28)

Since the coefficients are only C^{α}, we can expect to find only a weak solution u ∈ C^{1,α}(\overline{Ω^{+}(ψ)}) of (5.25)–(5.28) in the sense that u(x) satisfies (5.28) and, for any w ∈ C_{c}^{1}(R^{n}),

∫_{Ω^{+}(ψ)} ∑_{i,j=1}^{n} a_{ij}^{(ψ)}u_{x_{j}}w_{x_{i}} dx + ∫_{S_{ψ}} g_{ψ} w dH^{n−1}  (5.29)

− ∫_{∂_{l}Ω^{+}(ψ)} ρ̃((q^{+})^{2})q^{+}ν · e_{n} w dH^{n−1} = 0.

We will determine the iteration map J(ψ) = φ by solving (5.25)–(5.28) for u, extending u from Ω^{+}(ψ) to Ω so that u + q^{+}x_{n} ∈ K_{M}, and defining φ = u + q^{+}x_{n}. A fixed point of this map is obviously a solution of Problem (TFB) in Ω. In Sections 6–7, we prove the existence of such a map J, as well as its fixed point.

6. Fixed Boundary Problems in a bounded domain Ω_{R}^{+}(ψ)

In order to find a solution of (5.25)–(5.28) in the unbounded domain Ω^{+}(ψ), we first solve the corresponding problem in the bounded domain

Ω_{R}^{+}(ψ) := Ω^{+}(ψ) ∩ {x_{n} < R}, R > L,

and then pass to the limit as R → ∞. Thus we consider the following problem:

∑_{i,j=1}^{n}(a_{ij}^{(ψ)}u_{x_{j}})_{x_{i}} = 0  in Ω_{R}^{+}(ψ),  (6.1)

∑_{i,j=1}^{n} a_{ij}^{(ψ)}u_{x_{j}}ν_{i} = g_{ψ}(x)  on S_{ψ},  (6.2)

∑_{i,j=1}^{n} a_{ij}^{(ψ)}u_{x_{j}}ν_{i} = −ρ̃((q^{+})^{2})q^{+}ν · e_{n}  on ∂_{l}Ω_{R}^{+}(ψ),  (6.3)
u = 0  on ∂Ω_{R}^{+}(ψ) ∩ {x_{n} = R},  (6.4)

where a_{ij}^{(ψ)} are defined by (5.21) and ∂_{l}Ω_{R}^{+}(ψ) = ∂_{l}Ω^{+} ∩ ∂Ω_{R}^{+}(ψ).

We first note several properties of g_{ψ} which will be used later.

Lemma 6.1. There exists C > 0 depending only on the data and independent of M and R such that

∥g_{ψ}∥_{0,α,Ω_{1}} ≤ Cσ.  (6.5)

This is proved in [5, (4.48), page 334]. Another property of g_{ψ} is

Lemma 6.2.

∫_{S_{f}} g_{ψ}dH^{n−1} = ρ̃((q^{+})^{2})q^{+} ∫_{∂_{l}Ω_{R}^{+}} ν · e_{n}dH^{n−1}.

Proof. Note that

0 = ∫_{Ω_{R}} Δ(x_{n}) = − ∫_{Λ×{x_{n}=R}} e_{n} · νdx′ − ∫_{S_{f}} e_{n} · νdH^{n−1} − ∫_{∂_{l}Ω_{R}^{+}} e_{n} · νdH^{n−1},

where ν is the inward unit normal. Since ν = −e_{n} on Λ × {x_{n} = R}, then, from the last equality, we get

∫_{S_{f}} e_{n} · νdH^{n−1} + ∫_{∂_{l}Ω_{R}^{+}} e_{n} · νdH^{n−1} = |Λ|.  (6.6)

We also note that

∫_{S_{f}} G_{ψ}dH^{n−1} = ∫_{∂_{o}Ω} ρ(|Dφ^{−}|^{2})Dφ^{−} · νdH^{n−1},  (6.7)

since G_{ψ} is defined by (5.8) and φ^{−} ∈ C^{1,α}(\overline{Ω \ Ω_{R}^{+}(ψ)}) is a weak solution of (1.1) in Ω \ \overline{Ω_{R}^{+}(ψ)} and satisfies the boundary condition (2.19).

Then we have

∫_{S_{f}} g_{ψ}dH^{n−1} − ρ̃((q^{+})^{2})q^{+} ∫_{∂_{l}Ω_{R}^{+}} ν · e_{n}dH^{n−1}

= ∫_{S_{f}} G_{ψ}dH^{n−1} − ρ̃((q^{+})^{2})q^{+} (∫_{S_{f}} ν · e_{n}dH^{n−1} + ∫_{∂_{l}Ω_{R}^{+}} ν · e_{n}dH^{n−1})

= ∫_{S_{f}} G_{ψ}dH^{n−1} − ρ̃((q^{+})^{2})q^{+}|Λ| = 0,

where we used (6.7) and the fact that q^{+} satisfies (2.26) and (2.27). □

6.1. Existence of Solutions and Their Uniform Estimates Independent of R for the Linear Fixed Boundary Problem in Ω_{R}^{+}(ψ). In this section, we fix ψ ∈ K_{M} and write Ω_{R}^{+} for Ω_{R}^{+}(ψ) and a_{ij} for a_{ij}^{(ψ)} to simplify the presentation. Also, C denotes a universal constant depending only on the data and independent of M and R, which may be different at each occurrence. Furthermore, there exists σ_{0} > 0 depending only on M such that, when σ ≤ σ_{0}, S_{ψ} ⊂ {−1/10 < x_{n} < 1/10} by (5.5); we always assume σ ≤ σ_{0} below.

Lemma 6.3. Let σ ∈ (0, σ_{0}) be sufficiently small, depending only on the data and M. Let u ∈ H^{1}(Ω_{R}^{+}), R > L, be a weak solution of (6.1)–(6.4) in the sense that

(i) u = 0 on ∂Ω_{R}^{+} ∩ {x_{n} = R} as the trace;

(ii) for any w ∈ H^{1}(Ω_{R}^{+}) satisfying w = 0 on ∂Ω_{R}^{+} ∩ {x_{n} = R} in the sense of traces,
∫_{Ω_{R}^{+}} ∑_{i,j=1}^{n} a_{ij}u_{x_{j}}w_{x_{i}} dx + ∫_{S_{f}} g_{ψ} w dH^{n−1} − ∫_{∂_{l}Ω_{R}^{+}} ρ̃((q^{+})^{2})q^{+}ν · e_{n} w dH^{n−1} = 0.  (6.8)

Then

∥Du∥_{L^{2}(Ω_{R}^{+})} ≤ Cσ.  (6.9)

Proof. We use the approach in the proof of [5, Lemma 4.2]. We first choose w = u in (6.8) to obtain

∫_{Ω_{R}^{+}} ∑_{i,j=1}^{n} a_{ij}u_{x_{j}}u_{x_{i}} dx = − ∫_{S_{f}} g_{ψ}u dH^{n−1} + ∫_{∂_{l}Ω_{R}^{+}} ρ̃((q^{+})^{2})q^{+}ν · e_{n} u dH^{n−1}.

Now, if Q is a constant which will be chosen below, we use Lemma 6.2 to obtain

∫_{Ω_{R}^{+}} ∑_{i,j=1}^{n} a_{ij}u_{x_{j}}u_{x_{i}} dx = − ∫_{S_{f}} g_{ψ}(u − Q) dH^{n−1} − ∫_{∂_{l}Ω_{L}^{+}} ρ̃((q^{+})^{2})q^{+}ν · e_{n} (u − Q) dH^{n−1},

where we used that ν · e_{n} = 0 on ∂_{l}Ω_{R}^{+} \ ∂_{l}Ω_{L}^{+} by (2.14) and (2.17).

Choosing Q = (u)_{L} := \frac{1}{|Ω_{L}^{+}(ψ)|} ∫_{Ω_{L}^{+}(ψ)} u(x)dx and using the L^{2} estimate of the boundary

traces of functions in the Sobolev space H^{1}(Ω_{L}^{+}(ψ)), (6.5), and |ν · e_{n}| ≤ Cσ by (2.16), we obtain

∫_{Ω_{R}^{+}} ∑_{i,j=1}^{n} a_{ij}u_{x_{j}}u_{x_{i}} dx ≤ (∫_{S_{f}} g_{ψ}^{2}dH^{n−1})^{1/2} (∫_{S_{f}} (u − (u)_{L})^{2}dH^{n−1})^{1/2}

+ρ̃((q^{+})^{2})q^{+} (∫_{∂_{l}Ω_{L}^{+}} |ν · e_{n}|^{2}dH^{n−1})^{1/2} (∫_{∂_{l}Ω_{L}^{+}}(u − (u)_{L})^{2}dH^{n−1})^{1/2}

≤ Cσ (∫_{Ω_{L}^{+}(ψ)} ((u − (u)_{L})^{2} + |Du|^{2}) dx)^{1/2}

≤ Cσ (∫_{Ω_{L}^{+}(ψ)} |Du|^{2}dx)^{1/2} ,

where, in the last estimate, we used the Poincaré inequality in the domain Ω_{L}^{+}(ψ), and thus the constant C depends only on n, L, the norms in (2.15) and (2.16), and ∥f_{ψ}∥_{1,α,Λ}. On the other hand, if σ < \frac{1}{M}, then ∥f_{ψ}∥_{1,α,Λ} ≤ C with C > 0 independent of M, R, and ψ ∈ K_{M}. Now, using (3.35), we get

∫_{Ω_{R}^{+}} |Du|^{2}dx ≤ \frac{1}{λ} ∫_{Ω_{R}^{+}} ∑_{i,j=1}^{n} a_{ij}u_{x_{j}}u_{x_{i}} dx ≤ \frac{C}{λ} σ (∫_{Ω_{R}^{+}} |Du|^{2}dx)^{1/2} .

This completes the proof. □

Since ∂_{l}Ω ∩ ∂Ω_{R}^{+} is C^{1,α}, then, by [36], the weak solution u of (6.1)–(6.4) is in C^{1,α} (\overline{Ω}_{R}^{+} \ (\overline{S_{ψ}} ∪ \{x_{n} = R\})).

Lemma 6.4. Let σ and u be as in Lemma 6.3. Then, for any x_{n} ∈ (L, R), there exists x′ ∈ \overline{Λ} such that u(x′, x_{n}) = 0.

Proof. We follow the scheme of the proof [5, Lemma 4.3, Step 2]. Fix z ∈ (L, R). Suppose that there is no x′ ∈ \overline{Λ} such that u(x′, z) = 0. Then we can assume u(x′, z) > 0 for all x′ ∈ \overline{Λ}, since the case u(x′, z) < 0 for all x′ ∈ \overline{Λ} can be handled similarly. From the compactness of Λ and continuity of u, there exists a constant κ > 0 such that

u(x′, z) ≥ κ > 0 for all x′ ∈ \overline{Λ}.

Consider domain D := Λ × (z, R). Then u ∈ C(\overline{D}) ∩ C^{2}(\overline{D} \ W), with W = ∂_{l}Ω ∩ \{x_{n} = R\}, and u is a weak solution of

∑_{i,j=1}^{n}(a_{ij}u_{x_{j}})_{x_{i}} = 0 in D,

∑_{i,j=1}^{n} a_{ij}u_{x_{j}}ν_{i} = 0 on ∂_{l}D = ∂D ∩ ∂_{l}Ω ∩ {x_{n} < R},

u ≥ κ > 0 on ∂D ∩ {x_{n} = z}, u = 0 on ∂D ∩ {x_{n} = R}.

Thus, by the maximum principle, u ≥ 0 in D. In particular,

u_{ν} = −u_{x_{n}} ≥ 0  on (∂D ∩ {x_{n} = R}) \ W.  (6.10)

By the strong maximum principle [21, Theorem 8.19], u > 0 in the interior of D. Also, denoting D_{δ} = {x ∈ D : dist(x, ∂_{l}Ω) > δ}, where δ > 0 is fixed to be small so that D_{δ} ≠ ∅, we have

a_{ij} ∈ C^{0,α}(\overline{D_{δ/2}}), u ∈ C^{1,α}(\overline{D_{δ/2}}).

Thus, by [5, Lemma A.1],

u_{ν} = −u_{x_{n}} > 0  on \overline{D}_{δ} ∩ \{x_{n} = R\}.  (6.11)

Since u = 0 on Λ × {x_{n} = R} and a_{nn} ≥ λ > 0 from the ellipticity, we conclude from (6.10) and (6.11) that

∑_{i,j=1}^{n} a_{ij}u_{x_{j}}ν_{i} = −a_{nn}u_{x_{n}} \{ \begin{matrix}≥ 0  on (∂D ∩ \{x_{n} = R\}) \ W, \\ > 0  on \overline{D}_{δ} ∩ \{x_{n} = R\}. \end{matrix}

Since a_{nn} and u_{x_{n}} are continuous on \overline{D}_{δ} ∩ \{x_{n} = R\}, we conclude

∫_{\overline{Ω}∩\{x_{n}=R\}} ∑_{i,j=1}^{n} a_{ij}u_{x_{j}}ν_{i} dH^{n−1} > 0.  (6.12)

On the other hand, from (6.1)–(6.3),

0 = ∫_{Ω_{R}^{+}} ∑_{i,j=1}^{n}(a_{ij}^{ψ}u_{x_{j}})_{x_{i}}dx

= (∫_{S_{ψ}} + ∫_{∂_{l}Ω_{R}^{+}} + ∫_{\overline{Ω}∩\{x_{n}=R\}}) ∑_{i,j=1}^{n} a_{ij}u_{x_{j}}ν_{i}dH^{n−1}

= ∫_{S_{ψ}} g_{ψ}dH^{n−1} − ∫_{∂_{l}Ω_{R}^{+}} ρ̃((q^{+})^{2})q^{+}ν · e_{n}dH^{n−1} + ∫_{\overline{Ω}∩\{x_{n}=R\}} ∑_{i,j=1}^{n} a_{ij}u_{x_{j}}ν_{i}dH^{n−1}

= ∫_{\overline{Ω}∩\{x_{n}=R\}} ∑_{i,j=1}^{n} a_{ij}u_{x_{j}}ν_{i} dH^{n−1},

where the last equality follows from Lemma 6.2. Thus we conclude a contradiction with (6.12), which completes the proof. □

From (2.14)–(2.17), there exist r > 0 and C > 0 with the following properties: For any x_{0} ∈ ∂_{l}Ω∩{x_{n} > −1/2}, there exists an orthonormal coordinate system (y_{1}, …, y_{n}) with y_{n} = x_{n} and a function ζ : R^{n−1} → R such that

Ω ∩ B_{10r}(x_{0}) = {y_{1} > ζ(y_{2}, …, y_{n})} ∩ B_{10r}(x_{0}), ∥ζ∥_{1,α,R^{n−1}} ≤ C, Dζ(x_{0}) = 0. (6.13)

Proposition 6.5. Let u(x) be as in Lemma 6.3. If σ is sufficiently small, depending only on the data and M, then, for any x^{0} = (x_{1}^{0}, …, x_{n}^{0}) ∈ \overline{Ω_{R−10r}^{+}} with R − 10r > x_{n}^{0} ≥ L + 1,

∥Du∥_{0,α,B_{r}(x_{0})∩Ω_{R}^{+}} ≤ C∥Du∥_{L^{2}(B_{4r}(x_{0})∩Ω_{R}^{+})}.  (6.14)

Proof. We use the fact that u − K also satisfies equation (6.1) and the conormal boundary conditions (6.2) and (6.3) for any constant K ∈ R.

When x_{0} satisfies B_{2r}(x_{0}) ⊂ Ω_{R}^{+}, we apply [26, Theorem 3.13] to u − K and get

∥Du∥_{0,α,B_{r}(x_{0})∩Ω_{R}^{+}} ≤ C∥u − K∥_{L^{2}(B_{4r}(x_{0})∩Ω_{R}^{+})}.  (6.15)

When x_{0} ∈ ∂_{l}Ω^{+} with R − 10r > x_{n}^{0} ≥ L + 1, then, since the right-hand side of (6.3) vanishes on ∂_{l}Ω^{+} ∩ {x_{n} > L}, we use (5.24) and (a linear version of) the estimates for the conormal derivative problem [36], applied to u − K, to obtain

∥Du∥_{0,α,B_{r}(x_{0})∩Ω_{R}^{+}} ≤ C∥u − K∥_{C^{0}(B_{2r}(x_{0})∩Ω_{R}^{+})}.  (6.16)

Now we use the standard estimates [21, Theorem 8.17] for the equations of divergence form (extended to the case of local estimates near the boundary for the conormal problem, see e.g. [38, Chapter 4, Section 10] where these modifications are indicated in the parabolic case; note that the Lipschitz regularity of the boundary in our case allows to use the weak form of the conormal problem) to obtain

∥u − K∥_{C^{0}(B_{2r}(x_{0})∩Ω_{R}^{+})} ≤ C∥u − K∥_{L^{2}(B_{4r}(x_{0})∩Ω_{R}^{+})}.

That is, (6.15) is satisfied for the case x_{0} ∈ ∂_{l}Ω_{R}^{+}.

Now choosing K = \frac{1}{|B_{4r}(x_{0}) ∩ Ω_{R}^{+}|} ∫_{B_{4r}(x_{0})∩Ω_{R}^{+}} u dx, using the Poincare inequality, and noting

that the constant in the Poincare inequality can be chosen independent of x_{0} ∈ Ω_{R}^{+} by (6.13), we complete the proof. □

Proposition 6.6. Let σ and u(x) be as in Lemma 6.3. Then

∥u∥_{C^{0}(\overline{Ω_{R}^{+}})} ≤ Cσ.

Proof. Lemma 6.3 and Proposition 6.5 imply that

|Du| ≤ Cσ on Ω_{R}^{+} ∩ {L + 1 < x_{n} < R − 10r}.

Combining this with Lemma 6.4, we have

|u| ≤ Cσ on Ω_{R}^{+} ∩ {x_{n} ∈ (L + 1, R − 10r)}.

To extend this bound to the domains Ω_{R}^{+} ∩ {R − 10r < x_{n} < R} = Λ × (R − 10r, R) and Ω_{L}^{+}, we note that these domains are of the fixed size and structure (cylinder and a small perturbation cylinder of the form Λ × (a, b), respectively). Note that, on the boundary parts {x_{n} = L} ⊂ ∂Ω_{L}^{+} and {x_{n} = R−10r} ⊂ ∂(Λ×(R−10r, R)), we have |u| ≤ Cσ, as we proved above. Moreover, the right-hand sides of (6.2) and (6.3) are estimated in L^{∞} by Cσ from (2.16) and (6.5), respectively. Thus we can use the standard estimates [21, Theorem 8.15] for the equations of divergence form (extended to the case when we have the conormal boundary conditions on a part of the boundary, see e.g. [38, Chapter 4, Section 10]). □

Proposition 6.7. If σ is sufficiently small, there exists a unique weak solution u ∈ H^{1}(Ω_{R}^{+}) of (6.1)–(6.4). Furthermore, u ∈ C^{1,α}(\overline{Ω_{R−10r}^{+}}) satisfies (6.9) and

∥u∥_{1,α,Ω_{R−10r}^{+}} ≤ Cσ.  (6.17)

Proof. Since the map Ψ in (2.14) satisfies (2.16), then, if σ is sufficiently small, there exists the inverse map Φ : R^{n} → R^{n} of Ψ. The map Φ satisfies

∥Φ − Id∥_{2,α,R^{n}} ≤ Cσ.  (6.18)

Then, from (2.14),

Φ(∂_{l}Ω) ⊂ ∂Λ × (−∞, ∞).  (6.19)

Note that, if σ is small, then, by (2.16), (4.5), and (5.4), we conclude that the set Φ(Ω^{+}(ψ)) = {ψ ∘ Ψ < φ^{−} ∘ Ψ} ∩ (Λ × R) has the form

Φ(Ω^{+}(ψ)) = {x_{n} > f̃(x′)} ∩ (Λ × R) with ∥f̃∥_{1,α,R^{n−1}} ≤ CMσ.  (6.20)

Now we consider a map

Φ̃ = Φ^{1} ∘ Φ : R^{n} → R^{n},  for Φ^{1}(x′, x_{n}) = (x′, x_{n} − η(x_{n})f̃(x′)),  (6.21)

where η ∈ C^{∞}(R) is nonincreasing and satisfies η ≡ 1 on (−∞, 1/8) and η ≡ 0 on (1/4, ∞). From this definition, we have

∥Φ̃ − Id∥_{1,α,R^{n}} ≤ CMσ.

Thus, for small Mσ, the map Φ̃ is invertible and

∥Φ̃^{−1} − Id∥_{1,α,R^{n}} ≤ CMσ.

In particular, for small Mσ

∥Φ̃ − Id∥_{1,α,R^{n}} ≤ \frac{1}{10},  ∥Φ̃^{−1} − Id∥_{1,α,R^{n}} ≤ \frac{1}{10}.  (6.22)

Moreover, if Mσ is small, then

Λ × (0, ρ) = Φ̃ (Ω_{R}^{+} ∩ {x_{n} < ρ})  for ρ > 1/4.  (6.23)

Define the function v(y) := u(x) = u(Φ̃^{−1}(y)) for y ∈ C_{R} := Λ × (0, R). Then u is a weak solution of the conormal problem (6.1)–(6.4) if and only if v ∈ H^{1}(C_{R}) is a weak solution of the corresponding problem of the form (3.1)–(3.3) in C_{R}. To obtain the expressions of the coefficients a_{ij} and the right-hand sides f, g_{1}, and g_{2} of problem (3.1)–(3.2) for v, in terms of the data of problem (6.1)–(6.3) and the map Φ̃, we make the change of variables y = Φ̃(x) in (6.8). Then we get

∥a_{ij} − κ_{i}δ_{i}^{j}∥_{0,α,C_{R}} ≤ CMσ for i, j = 1, …, n, (6.24)

with κ_{i} defined by (5.23), where we used (5.24) and (6.22). In this case, the right-hand side in (3.1) is f = 0; to estimate g_{1} and g_{2} in (3.2), we use (2.16), (6.5), (6.22) to obtain

∥g_{k}∥_{0,α,Σ_{k}} ≤ Cσ,  k = 1, 2.  (6.25)

Also, from (6.4) and (6.23), we see that h = 0 in (3.3) for v.

Now the existence and uniqueness of v(y) follows from Theorem 3.1. Thus u(x) := v(Φ̃(x)) is a unique weak solution of (6.1)–(6.4). Estimate (6.9) follows from Lemma 6.3.

It remains to prove (6.17). For any x ∈ \overline{Ω_{R}^{+}} ∩ \{1 ≤ x_{n} ≤ R − 10r\}, the standard interior estimates [21, Theorem 8.32] and the estimates for oblique derivative problems [36] imply

∥u∥_{1,α,B_{r}(x_{0})∩Ω_{R}^{+}} ≤ C(∥u∥_{C^{0}(B_{4r}(x_{0})∩Ω_{R}^{+})} + ∥ρ̃((q^{+})^{2})q^{+}ν · e_{n}∥_{0,α,B_{r}(x_{0})∩∂Ω_{R}^{+}}) ≤ Cσ, (6.26)

where we used Proposition 6.6 and (2.16). Thus it suffices to obtain an estimate in Ω_{R}^{+}∩{x_{n} ≤ 1}. By (6.22) and (6.23), it is sufficient to show that

∥v∥_{1,α,Λ×(0,1)} ≤ Cσ,  (6.27)

since the similar estimate then holds for ∥u∥_{1,α,Ω_{R}^{+}∩{x_{n}≤1}}.

To show (6.27), we note that using (6.22), (6.23), and Proposition 6.6 yields

∥v∥_{L^{2}(Λ×(0,2))} ≤ 2∥u∥_{L^{2}(Ω_{R}^{+}∩{x_{n}<2})} ≤ Cσ.

Now estimate (3.8), applied to v in the cylinder Λ × (0, 2), with the use of (6.25), implies (6.27), if σ is small depending only on M and the data. □

7. Existence of Solutions of the Free Boundary Problem

We now show the existence of a solution of the free boundary problem.

7.1. Solutions of the Fixed Boundary Problem in the Infinite Nozzle. We first have

Proposition 7.1. If σ is sufficiently small, then there exists a unique weak solution u ∈ C^{1,α}(\overline{Ω^{+}(ψ)}) of (5.25)–(5.28). This solution satisfies

∥Du∥_{L^{2}(Ω^{+}(ψ))} ≤ Cσ,  (7.1)
∥u∥_{1,α,Ω^{+}(ψ)} ≤ Cσ,  (7.2)
lim_{ρ→∞} ∥u∥_{1,α,Ω∩{x_{n}>ρ}} = 0.  (7.3)

Moreover, we have (i) u = lim_{R→∞} u_{R}, where u_{R} is the unique solution of (6.1)–(6.4) for each R > 0 and the

convergence is in C^{1,β} on any compact subset of \overline{Ω^{+}(ψ)} for any 0 ≤ β < α; (ii) For every R ≥ L, there exists x′ ∈ \overline{Λ} such that u(x′, R) = 0.

Proof. We first show that any weak solution u ∈ C^{1,α}(\overline{Ω^{+}(ψ)}) of (5.25)–(5.28) satisfies (7.3). From (5.28), for any ε > 0, there exists ρ > L such that

∥u∥_{C(Ω∩{x_{n}>ρ})} ≤ ε.

Note that (2.17) implies that the right-hand side of (5.27) vanishes on ∂_{l}Ω^{+}(ψ) ∩ {x_{n} > ρ}. Then the standard estimates [21, Theorem 6.2, Lemma 6.29] imply that, for x = (x′, x_{n}) with x_{n} > L + 2r,

∥u∥_{1,α,B_{r}(x)∩Ω} ≤ C∥u∥_{0,0,B_{2r}(x)∩Ω} ≤ ε.

Thus, (7.3) follows.

Now we prove the uniqueness of weak solutions of (5.25)–(5.28) in C^{1,α}(\overline{Ω^{+}(ψ)}). Let u, v ∈ C^{1,α}(\overline{Ω^{+}(ψ)}) be two solutions of (5.25)–(5.28). Then, for any R > L + diam(Λ), define

ζ(x) := η_{R}(x)(u(x) − v(x)),

where η_{R}(x) = η(|x|/R) and η ∈ C^{∞}(\overline{R_{+}}) is a nonnegative function such that η ≡ 1 on [0, 1]. Then ζ ∈ C_{c}^{1}(R^{n} ∩ \overline{Ω^{+}}). Using ζ as a test function in (5.29) for u and v respectively and subtracting them, we get

0 = ∫_{Ω^{+}} ∑_{i,j=1}^{n} a_{ij}^{(ψ)} ((u_{x_{i}} − v_{x_{i}})(u_{x_{j}} − v_{x_{j}})η_{R} + (u − v)(u_{x_{j}} − v_{x_{j}})∂_{x_{i}}η_{R}) dx.

Thus,

∫_{Ω^{+}} ∑_{i,j=1}^{n} a_{ij}^{(ψ)}(u_{x_{i}} − v_{x_{i}})(u_{x_{j}} − v_{x_{j}})η_{R}dx ≤ \frac{C}{R} ∫_{Ω^{+}∩(B_{2R}(0)\B_{R}(0))}(|Du| + |Dv|)(|u| + |v|)dx.

Note that |Ω^{+} ∩ (B_{2R}(0) \ B_{R}(0))| ≤ 2R|Λ|. Thus, using the ellipticity of a_{ij}^{(ψ)}, we get

λ ∫_{Ω^{+}∩B_{R}(0)} |Du − Dv|^{2}dx ≤ C(∥u∥_{1,0,Ω\B_{R}(0)} + ∥v∥_{1,0,Ω\B_{R}(0)})

for large R. Now we send R → ∞ and use (7.3) to get

λ ∫_{Ω_{R}^{+}} |Du − Dv|^{2}dx = 0.

Thus, u − v = const., which yields u = v by using (7.3).

It remains to prove the existence of a weak solution of (5.25)–(5.28) satisfying (7.1), (7.2), and properties (i)–(ii) of Proposition 7.1.

The existence of a unique solution u_{R} ∈ C^{1,α}(\overline{Ω_{R−10r}^{+}}) ∩ H^{1}(Ω_{R}^{+}) of (6.1)–(6.4) for any R > L follows from Proposition 6.7.

Now, let R_{j} → ∞. Using estimates (6.9) and (6.17) for u_{R}, which hold by Proposition 6.7, we can extract from \{u_{R_{j}}\} a subsequence converging in C^{1,α/2} on compact subsets of \overline{Ω} to a function u ∈ C^{1,α}(\overline{Ω}) that satisfies (7.1). Indeed, this can be achieved by extracting a subsequence converging in C^{1,α/2}(\overline{Ω_{L+1}^{+}}), a further subsequence converging in C^{1,α/2}(\overline{Ω_{L+2}^{+}}), etc., and by using the diagonal procedure. Obviously, u is a weak solution of (5.25)–(5.27), since we can pass to the limit in (5.29) for any fixed w ∈ C_{c}^{1}(R^{n} ∩ \overline{Ω^{+}}).

Moreover, this solution u satisfies property (ii): Indeed, for fixed R_{0} ≥ 1, R_{j} ≥ R_{0} for sufficiently large j. For such j, by Lemma 6.4, there exists x′_{j} ∈ \overline{Λ} such that u_{R_{j}}(x′_{j}, R_{0}) = 0. Then there exists a subsequence of \{x′_{j}\} converging to some point x′ ∈ \overline{Λ}. Since u_{R_{j}}(·, R_{0}) → u(·, R_{0}) uniformly in \overline{Λ}, we have u(x′, R_{0}) = 0. Thus, the solution u satisfies (i) and (ii).

Also, since each u_{R} satisfies (6.9) and (6.17) with C independent of R, then (7.1) and (7.2) hold.

Now we prove that u satisfies (5.28). From (7.1), for any ε > 0, there exists ρ > L such that

∥Du∥_{L^{2}(Ω^{+}∩{x_{n}>ρ})} ≤ ε.

Then, by Proposition 6.5 (which holds for u as u_{R}),

∥Du∥_{0,α,Ω^{+}∩{x_{n}>ρ+1}} ≤ Cε.

From this estimate and property (ii) proved above, we have

∥u∥_{C^{0}(Ω^{+}∩{x_{n}>ρ+1})} ≤ Cε.

Now (5.28) is proved. □

7.2. Iteration Map. We first define an extension operator P_{ψ}, similar to [4, Proposition 4.5] and [5, Proposition 5.1], to obtain

Proposition 7.2. Let σ > 0 be sufficiently small and φ^{−} satisfy (4.5). Then, for any ψ ∈ K_{M}, there exists an extension operator P_{ψ} : C^{1,α}(\overline{Ω^{+}(ψ)}) → C^{1,α}(\overline{Ω}) satisfying the following two properties:

(i) There exists C depending only on n, γ, q_{0}^{+}, and α (but independent of M, R, σ, and ψ) such that, for the solution u ∈ C^{1,α}(\overline{Ω^{+}(ψ)}) of problem (5.25)–(5.28) and φ = u + q^{+}x_{n},

∥P_{ψ}φ − q^{+}x_{n}∥_{1,α,Ω} ≤ Cσ;  (7.4)

(ii) Let β ∈ (0, α). Let a sequence ψ_{k} ∈ K_{M} converges to ψ ∈ C^{1,β}(\overline{Ω}) in the norm ∥ · ∥_{1,β,Ω}^{(−1)} defined in (5.2). Then ψ ∈ K_{M}. Let u_{k} ∈ C^{1,α}(\overline{Ω^{+}(ψ_{k})}) and u ∈ C^{1,α}(\overline{Ω^{+}(ψ)}) be the solutions of problems (5.25)–(5.28) for ψ_{k} and ψ, respectively, and let φ_{k} = u_{k} + q^{+}x_{n} and φ = u + q^{+}x_{n}. Then P_{ψ_{k}}φ_{k} → P_{ψ}φ in the norm ∥ · ∥_{1,β,Ω}^{(−1)}.

Proof. Let ψ ∈ K_{M}. We first define an appropriate extension operator P_{ψ} : C^{1,α}(\overline{Ω^{+}(ψ)}) → C^{1,α}(\overline{Ω}).

Let φ ∈ C^{1,α}(\overline{Ω^{+}(ψ_{j})}) and u = φ − q^{+}x_{n}. Then v := u ∘ Φ̃^{−1} satisfies v ∈ C^{1,α}(\overline{C_{(0,∞)}}) with C_{(a,b)} = Λ × (a, b), where Φ̃ is defined by (6.21).

We first define the extension operator E_{1} : C^{1,β}(\overline{C_{(0,∞)}}) → C^{1,β}(\overline{C_{(−2,∞)}}) for any β ∈ (0, 1). Let v ∈ C^{1,β}(\overline{C_{(0,∞)}}). Define E_{1}v = v in C_{(0,∞)}. For (x′, x_{n}) ∈ C_{(−1,0)}, define

E_{1}v(x′, x_{n}) = ∑_{i=1}^{2} c_{i}v(x′, −\frac{x_{n}}{i}),

where c_{1} = −3 and c_{2} = 4, which are determined by ∑_{i=1}^{2} c_{i} (−\frac{1}{i})^{m} = 1 for m = 0, 1. Note that, by (6.22) and for sufficiently small σ, we have Ω ⊂ Φ̃(C_{(−2,∞)}). Thus, (E_{1}v) ∘ Φ̃ ∈ C^{1,α}(\overline{Ω}).

Finally, we define

P_{ψ}(φ)(x) = (E_{1}v) ∘ Φ̃(x) + q^{+}x_{n}.

Obviously, P_{ψ}(φ) ∈ C^{1,α}(\overline{Ω}). Also, since Φ̃(Ω^{+}(ψ)) = C_{(0,∞)}, we have

P_{ψ}(φ)(x) = v ∘ Φ̃(x) + q^{+}x_{n} = u(x) + q^{+}x_{n} = φ(x) for x ∈ Ω^{+}(ψ),

i.e., P_{ψ} : C^{1,α}(\overline{Ω^{+}(ψ)}) → C^{1,α}(\overline{Ω}) is an extension operator.

Now, using (6.22), the following estimate is obtained as in the proof of [4, Proposition 4.5]:

∥P_{ψ}φ − q^{+}x_{n}∥_{1,α,Ω} ≤ C∥φ − q^{+}x_{n}∥_{1,α,Ω^{+}(ψ)}.  (7.5)

Then, if u and φ are those defined in (i), we have (7.2) for u = φ − q^{+}x_{n} by Proposition 7.1, and thus (7.5) implies (7.4). The assertion (i) is proved.

Now we prove assertion (ii) of the proposition. First, by Lemma 5.1, ψ ∈ K_{M}.

By (7.4) applied to ψ_{j} and by Lemma 5.1, there exists a subsequence (still denoted by) φ_{j}(x) such that

P_{ψ_{j}}φ_{j} → φ̃ in C_{(−1)}^{1,β}(\overline{Ω})  (7.6)

for some φ̃ ∈ C^{2,α}(\overline{Ω}).

Denote ψ̃_{k} := ψ_{k} ∘ Ψ and ψ̃ := ψ ∘ Ψ. Then, by (2.16), ψ̃_{k} → ψ̃ in C^{1,β}(\overline{Λ} × [−1, 1]). Now denote by f̃_{k}(x′) the function from (6.20) for ψ_{k}(x). Then, by (2.16), (4.5), and (5.4), f̃_{k} → f̃ in C^{1,β}(\overline{Λ}). Now let Φ̃_{k} and Φ̃ be the maps from (6.21) for f̃_{k} and f̃, respectively. Then Φ̃_{k} and Φ̃ and their inverse maps satisfy (6.22), and Φ̃_{k} → Φ̃ and Φ̃_{k}^{−1} → Φ̃^{−1} in C^{1,β} on compact subsets of R^{n}. Note v_{k} := u_{k} ∘ Φ̃_{k}^{−1} ∈ C^{1,α}(\overline{Λ} × [0, ∞)). Recall that u_{k} is a weak solution of (5.25)–(5.28) in the domain Ω^{+}(ψ_{k}), with coefficients a_{ij}^{(ψ_{k})} defined by (5.22) for ψ_{k} and the corresponding right-hand sides (5.18) and (5.27) for ψ_{k}. Then, by Proposition 7.1, each u_{k} satisfies (7.1) and (7.2). Then

∥Dv_{k}∥_{L^{2}(Λ×(0,∞))} ≤ Cσ,  (7.7)
∥v_{k}∥_{1,α,Λ×(0,∞)} ≤ Cσ for k = 1, ….  (7.8)

Moreover, v_{k} is a weak solution of a conormal problem

∑_{i,j=1}^{n}(b_{ij}^{(ψ_{k})}∂_{x_{j}}v_{k})_{x_{i}} = 0  in Λ × (0, ∞),  (7.9)

∑_{i,j=1}^{n} b_{ij}^{(ψ_{k})}∂_{x_{j}}v_{k}ν_{i} = g̃_{ψ_{k}}(x)  on Λ × {0},  (7.10)

∑_{i,j=1}^{n} b_{ij}^{(ψ_{k})}∂_{x_{j}}v_{k}ν_{i} = h_{ψ_{k}}(x)  on ∂Λ × (0, ∞),  (7.11)

where b_{ij}^{(ψ_{k})}, g̃_{ψ_{k}}, and h_{ψ_{k}} are defined by the coefficients and right-hand sides of (5.25)–(5.27) for ψ_{k} and by Φ̃_{k}^{−1}. In particular, since a_{ij}^{(ψ_{k})} → a_{ij}^{(ψ)} in C^{1,β} on compact subsets of \overline{Ω} as k → ∞, which follows from (5.22), and since Φ̃_{k}^{−1} → Φ̃^{−1} in C^{1,β} on compact subsets of R^{n}), it is easy to show that, for any compact K ⊂ \overline{Λ} × [0, ∞),

b_{ij}^{(ψ_{k})} → b_{ij}^{(ψ)} in C^{1,β}(K)  and  (g̃_{ψ_{k}}, h_{ψ_{k}}) → (g̃_{ψ}, h_{ψ}) in C^{β}(K),

where b_{ij}^{(ψ)}, g̃_{ψ}, and h_{ψ} are the coefficients and right-hand sides of (7.9)–(7.11) for ψ. By (7.8), there exists a subsequence v_{k_{m}} which converges in C^{1,β} on the compact subsets of \overline{Λ} × [0, ∞) to some v ∈ C^{1,β}(\overline{Λ} × [0, ∞)). Then, choosing w ∈ C_{c}^{1}(R^{n}) in the weak formulation (5.29) for the conormal problem (7.9)–(7.11):

∫_{Λ×(0,∞)} ∑_{i,j=1}^{n} b_{ij}^{(ψ_{k})}v_{x_{k}}w_{x_{i}} dx + ∫_{Λ×{0}} g̃_{ψ_{k}}w dx′ − ∫_{∂Λ×(0,∞)} h_{ψ_{k}}w dH^{n−1} = 0.

We pass to the limit for the subsequence {k_{m}} and conclude that v is a weak solution of the problem (7.9)–(7.11) with coefficients b_{ij}^{(ψ)} and right-hand sides g̃_{ψ} and h_{ψ}. Note also that v satisfies (7.7) since each v_{k_{m}} satisfies (7.7). Transforming back, we conclude that ũ := v ∘ Φ̃ is a weak solution of the problem (5.25)–(5.27) for ψ, and ũ satisfies (7.1). Now the proof of Proposition 7.1 implies that ũ satisfies (5.28). Then the uniqueness assertion of Proposition 7.1 implies u = ũ on Ω^{+}(ψ). Thus

P_{ψ}(φ) = (E_{1}v) ∘ Φ̃ + q^{+}x_{n}.  (7.12)

Finally,

P_{ψ_{km}} (φ_{k_{m}}) = (E_{1}v_{k_{m}}) ∘ Φ̃_{k_{m}} + q^{+}x_{n} → (E_{1}v) ∘ Φ̃ + q^{+}x_{n} in C^{1,β}(K)

for any compact K ⊂ \overline{Ω}, since Φ̃_{k} → Φ̃, v_{k_{m}} → v in C^{1,β}(K), and E_{1} : C^{1,β}(\overline{C_{(0,∞)}}) → C^{1,β}(\overline{C_{(−2,∞)}}) is continuous. From this, using (7.12) and (7.6), we get P_{ψ_{km}} (φ_{k_{m}}) → P_{ψ}(φ) in C_{(−1)}^{1,β}(\overline{Ω}). Moreover, by the same argument, from any subsequence of P_{ψ_{k}}(φ_{k}), we can extract

a further subsequence converging in C_{(−1)}^{1,β}(\overline{Ω}) to the same limit P_{ψ}(φ). Thus the whole sequence P_{ψ_{k}}(φ_{k}) converges to this limit. Proposition 7.2 is proved. □

Now we can define the iteration map J : K_{M} → C^{1,α}(\overline{Ω}) by

Jψ := P_{ψ}φ  (7.13)

with φ(x) = u(x)+q^{+}x_{n}, where u ∈ C^{1,α}(\overline{Ω^{+}(ψ)}) is the unique solution of problem (5.25)–(5.28) with ψ ∈ K_{M}. By Proposition 7.2(ii), J is continuous in the C^{1,β}(\overline{Ω})-norm for any positive β < α.

7.3. Existence of Solutions of the Free Boundary Problem. Now we can prove the existence of solutions of the free boundary problem. We denote by φ both the function φ in Ω^{+}(ψ) and its extension P_{ψ}φ.

Choose M to be the constant C from (7.4). Then, for ψ ∈ K_{M}, we have from Proposition 7.2(i) that φ := Jψ ∈ K_{M} if σ > 0 is sufficiently small and depends only on n, γ, q^{+}, and Ω, since M > 0 is now fixed. Thus, (7.13) defines the iteration map J : K_{M} → K_{M} and, from Proposition 7.2(ii), J is continuous on K_{M} in the C_{(−1)}^{1,α/2}(\overline{Ω})-norm.

In order to find a solution of Problem (TFB), we seek a fixed point of the map J. We use the Schauder Fixed Point Theorem (cf. [21, Theorem 11.1]) in the following setting:

Let σ > 0 satisfy the conditions of Proposition 7.2. By Lemma 5.1, the set K_{M} is a compact convex subset of C_{(−1)}^{1,α/2}(\overline{Ω}). We have shown that J(K_{M}) ⊂ K_{M} and J is continuous in the

C_{(−1)}^{1,α/2}(\overline{Ω})-norm. Then, by the Schauder Fixed Point Theorem, J has a fixed point φ ∈ K_{M}. If φ is such a fixed point, then

φ̃(x) := min(φ^{−}(x), φ(x))

is a classical solution of Problem (TFB) and S_{φ} is its free boundary.

It follows that φ̃ is a solution of Problem (FB), thus Problem (TN), provided that σ is small enough so that (7.4) implies that |Dφ(x)| < c_{∗} − ε, where ε = (c_{∗} − q_{0}^{+})/2. Indeed, then (4.1) implies that φ lies in the untruncated region for equation (4.4). Also, |Dφ̃| < c_{∗} − ε on Ω^{+}(φ̃) := {φ̃(x) < φ^{−}(x)} and |Dφ̃| > c_{∗} on Ω \ Ω^{+} since φ̃ = φ on Ω^{+}(φ̃) and φ̃ = φ^{−} on Ω \ Ω^{+}.

8. Uniqueness and Stability

We now prove the uniqueness and stability of solutions of Problem (TN) when φ^{−} ∈ C^{2,α}. We first prove the following lemma.

Lemma 8.1. Let φ^{−} satisfy (4.5). Let φ ∈ C^{0,1}(Ω) be a solution of Problem (TN) with ω ∈ (0, c_{∗}) satisfying (2.24). Then there exists σ_{0} > 0 depending only on n, α, γ, and q_{0}^{+} such that, if σ ∈ (0, σ_{0}), φ satisfies (2.23), (2.30), and

∥Dφ − ωe_{n}∥_{C^{0}(Ω∩{x_{n}>R})} → 0  as R → ∞.  (8.1)

Proof. The proof that φ satisfies (2.23) and (2.30) is the same as in [5, Lemma 7.1].

Now we prove (8.1). Setting w(x) := φ(x) − ωx_{n} and using (2.17), (2.23), and (2.30), we see that w satisfies

∑_{i,j=1}^{n}(a_{ij}w_{x_{j}})_{x_{i}} = 0 in Ω ∩ {x_{n} > L},

∑_{i,j=1}^{n} a_{ij}w_{x_{j}}ν_{i} = −ρ̃((q^{+})^{2})q^{+}ν · e_{n} on ∂_{l}Ω^{+} ∩ {x_{n} > L},

lim_{R→∞} ∥w∥_{C(Ω)∩{x_{n}>R})} = 0,

where

a_{ij}(x) = ∫_{0}^{1} {ρ̃(|tDφ(x) + (1 − t)ωe_{n}|^{2}) δ_{i}^{j}
+2ρ̃′ (|tDφ(x) + (1 − t)ωe_{n}|^{2}) (tφ_{x_{i}}(x) + (1 − t)ωδ_{i}^{n}) (tφ_{x_{j}}(x) + (1 − t)ωδ_{j}^{n}) }dt.

By (2.24), the equation is elliptic if σ is small. Then the proof of (8.1) is similar to the proof of (7.3) in Proposition 7.1. □

Now we show that ω in Problem (TN) is uniquely determined by φ^{−}.

Proposition 8.2. There exists σ_{0} depending only on the data as in Lemma 8.1 such that, if φ^{−} satisfies (4.5) and φ ∈ C^{0,1}(Ω) is a solution of Problem (TN) with ω ∈ (0, c_{∗}) satisfying (2.24), then ω = q^{+} in (2.20), with q^{+} defined by (2.26) and φ(x) satisfies (2.28).

This can be achieved by following the proof of [5, Proposition 7.2] and using Lemma 8.1 instead of [5, Lemma 7.1].

It remains to prove the uniqueness of solutions of Problem (TN) satisfying (2.24).

Proposition 8.3. There exists σ_{0} depending only on the data as in Lemma 8.1 such that, if φ^{−} satisfies (4.5), then the solution φ ∈ C^{0,1}(Ω) of Problem (TN) with ω = q^{+} satisfying (2.24) is unique.

The rest of this section is to prove Proposition 8.3 by using a version of the partial hodograph transform developed in [6].

8.1. Extension of φ^{−} to the Domain Ω. Since φ^{−} satisfies (2.19) and (4.5) in the domain Ω_{1} := Ω ∩ {x_{n} < 1}, then we can extend φ^{−} to Ω so that the extension (still denoted by) φ^{−} is in C^{2,α}(Ω), coincides with the original φ^{−} in Ω ∩ {x_{n} < 1/2}, and

∥φ^{−} − φ_{0}^{−}∥_{2,α,Ω} ≤ C(n, α)σ,  (8.2)
supp(φ^{−} − φ_{0}^{−}) ⊂ Ω ∩ {x_{n} < L},  (8.3)
∂_{ν}φ^{−} = 0  on ∂_{l}Ω,  (8.4)

where φ_{0}^{−}(x) = q_{0}^{−}x_{n}. To construct such an extension, we first note that we can modify the mapping Ψ and replace it by a mapping Ψ_{1} ∈ C^{2,α}(R^{n}, R^{n}) such that Ψ_{1} satisfies (2.14), ∥Ψ_{1} − Id∥_{2,α,R^{n}} ≤ Cσ, (2.17), and

DΨ_{1}(x)ν_{∂C}(x) = ζ(x)ν_{∂_{l}Ω}(Ψ_{1}(x)) for any x ∈ ∂C,

where C = Λ×(−∞, ∞), ν_{∂C}(·) and ν_{∂_{l}Ω}(·) are the inward unit normals to ∂C and ∂_{l}Ω respectively, and ∥ζ − 1∥_{2,α,∂C} ≤ Cσ. To construct such Ψ_{1}, we first define the corresponding C^{2,α} mapping Ψ̃_{1} between sufficiently small neighborhoods (depending only on Λ and σ) of ∂C and ∂_{l}Ω by

Ψ̃_{1}(z + sν_{∂C}(z)) = Ψ(z) + sν_{∂_{l}Ω}(Ψ(z))

for z ∈ ∂C and s ∈ (0, ε), where ε > 0 depends only on Λ and L. Then we let

Ψ_{1}(x) = η_{ε}(d_{∂_{l}Ω}(x))Ψ̃_{1}(x) + (1 − η_{ε}(d_{∂_{l}Ω}(x)))Ψ(x),

where d_{∂_{l}Ω}(·) is the distance function to ∂_{l}Ω and η_{ε} : R → R is a smooth function satisfying 0 ≤ η_{ε} ≤ 1 on R, η_{ε} ≡ 1 on [−∞, 7ε/8), and η_{ε} ≡ 0 on (ε, ∞). Now we define φ̃^{−}(x) = φ^{−}(Ψ_{1}(x)) on Ψ_{1}^{−1}(Ω_{1}). Then, if σ is small, φ̃^{−} is defined at least on Λ×(−7/8, 7/8) and satisfies Dφ̃^{−}·ν_{C} = 0 on ∂Λ × (−7/8, 7/8). Now we define

φ̂^{−}(x) = η_{7/8}(x_{n})φ̃^{−}(x) + (1 − η_{7/8}(x_{n}))q^{−}x_{n} on Λ × (−7/8, ∞),

and finally the function φ^{−}(x) = φ̂^{−}(Ψ_{1}^{−1}(x)) is a C^{2,α} extension of φ^{−} satisfying (8.2)–(8.4).

Denote

g = div (ρ(|Dφ^{−}|^{2})Dφ^{−}) in Ω. (8.5)

Since φ^{−} satisfies (1.1) in Ω_{1}, then, from (8.2)–(8.3), we have that g satisfies

\begin{matrix}g ∈ C^{α}(\overline{Ω}),  ∥g∥_{0,α,Ω} ≤ Cσ, \\ g ≡ 0  on (Ω ∩ \{x_{n} < 1\}) ∪ (Ω ∩ \{x_{n} > L\}) . \end{matrix}  (8.6)

From now on, we use the extended function φ^{−} = φ^{−}(x), and C may denote a different constant at each occurrence, depending only on the data, unless otherwise is specified.

8.2. Partial Hodograph Transform. Let φ be a solution of Problem (TN) with ω = q^{+} and satisfy (2.24). Then Ω^{+} := {|Dφ| < c_{∗}} has the form (2.23) and (2.30). Define a function u in Ω^{+} by

u(x) := φ^{−}(x) − φ(x).

Then (2.31) and (8.2) imply

∥u − (q_{0}^{−} − q^{+})x_{n}∥_{1,α,Ω^{+}} ≤ Cσ.  (8.7)

In particular, if σ is sufficiently small, we use (2.29) to get

0 < (q_{0}^{−} − q_{0}^{+})/2 ≤ u_{x_{n}}(x) ≤ 2(q_{0}^{−} − q_{0}^{+})  for any x ∈ Ω^{+}.  (8.8)

Furthermore, (2.28) and (8.3) imply

∥u − (q_{0}^{−} − q^{+})x_{n}∥_{C^{1}(Ω∩{x_{n}>R})} → 0 as R → ∞.  (8.9)

From (2.24) with sufficiently small σ, φ satisfies (1.1) in Ω^{+} and (2.4) on S. Then, since S ⊂ Ω_{1} and φ^{−} satisfies (1.1) in Ω_{1} and (8.4) on ∂_{l}Ω, we see that u is a solution of the following problem:

div (A(x, Du)) = −g in Ω^{+},
A(x, Du) · ν = 0 on S,
A(x, Du) · ν = 0 on ∂_{l}Ω^{+},

(8.10)

where, for x ∈ Ω^{+} and P ∈ R^{n},

A(x, P) = ρ(|Dφ^{−}(x) − P |^{2})(Dφ^{−}(x) − P) − ρ(|Dφ^{−}(x)|^{2})Dφ^{−}(x).  (8.11)

The equation in (8.10) is uniformly elliptic for u if σ is sufficiently small, which follows from (4.5) and (8.7), since

0 < c_{0} ≤ Φ′(q) = ρ(q^{2}) + 2q^{2}ρ′(q^{2}) ≤ C for q near q_{0}^{+},

for some constants c_{0} and C > 0. Note that u satisfies the weak form of problem (8.10):

∫_{Ω^{+}}(A(x, Du) · Dη − gη) dx = 0  for any η ∈ C_{0}^{1}(R^{n}).  (8.12)

Since φ = φ^{−} on S, it follows that

u = 0  on S.  (8.13)

Now we make a change of variables z = Φ(x) to rewrite problem (8.12) into a cylindric domain C^{+} := Φ(Ω^{+}), where Ψ and its inverse Φ are the maps in (2.14), (2.16), and (6.18). It follows that, if σ is sufficiently small, then

C^{+} = (Λ × (−∞, ∞)) ∩ {z_{n} > f̃(z′)},  ∥f̃∥_{1,α,R^{n−1}} ≤ CMσ,  (8.14)

and ∂C^{+} = \overline{∂_{l}C^{+}} ∪ S̃, where

∂_{l}C^{+} := (∂Λ × (−∞, ∞)) ∩ {z_{n} > f̃(z′)} = Φ(∂_{l}Ω^{+}),
S̃ := (Λ × (−∞, ∞)) ∩ {z_{n} = f̃(z′)} = Φ(S).

Also, Ω^{+} = Ψ(C^{+}). Consider the function

ũ(z) = u(Ψ(z)) for z ∈ C^{+}.

Then (2.16), (2.17), and (8.7) imply

∥ũ − (q_{0}^{−} − q^{+})x_{n}∥_{1,α,C^{+}} ≤ Cσ.  (8.15)

In particular, ũ satisfies (8.8) if σ is small. By (8.9) and (2.17),

∥ũ − (q_{0}^{−} − q^{+})x_{n}∥_{C^{1}(Λ×(R,∞))} → 0  as R → ∞.  (8.16)

From (8.13), ũ satisfies

ũ = 0 on S. (8.17)

Moreover, ũ is a weak solution of a fixed boundary problem of the form (8.10) in the domain C^{+} with modified functions Ã(x, P) and g̃(x) satisfying Ã ∈ C^{1}(\overline{Λ × (−1, ∞)} × R^{n}) and g̃ ∈ C^{α}(\overline{Λ × (−1, ∞)}). Precisely, ũ satisfies

∫_{C^{+}}(Ã(x, Dũ) · Dη − g̃η) dx = 0  for any η ∈ C_{0}^{1}(R^{n}).  (8.18)

This is obtained by using the test function η ∘ Ψ, making the change of variables x → Ψ(x) in (8.12), and obtaining the expressions for Ã in C^{+} × R^{n} and g̃ in C^{+} in terms of A, g, and the map Ψ which are well defined in Λ × (−1, ∞) × R^{n} and Λ × (−1, ∞), respectively. We do not write the explicit expressions of Ã and g̃, but only point out some properties of Ã and g̃, which can be obtained readily by using these expressions. First, we emphasize that Ã and g̃ are given in terms of the original functions A, g, and the map Ψ, which are independent of a solution φ of Problem (TN). Next, using (2.17), we see that, for any R > 0, there exists C_{R} > such that

|D_{P} A(Ψ(x), P) − D_{P} Ã(x, P)| ≤ C_{R}∥Ψ∥_{C^{1}(R^{n})}  for any x ∈ C^{+}, P ∈ R^{n}, |P | < R. (8.19)

Furthermore, from (2.17), it follows that A = Ã for x_{n} > L, i.e.,

A(x, P) = Ã(x, P)  for any x ∈ Λ × (L, ∞), P ∈ R^{n}.  (8.20)

Similarly, from (2.16), (2.17), and (8.6), we conclude that g̃ satisfies

\begin{matrix}g̃ ∈ C^{α}(\overline{Λ} × [−1, ∞)), ∥g̃∥_{0,α,Λ×(−1,∞)} ≤ Cσ, \\ g̃ ≡ 0 on (Ω ∩ \{x_{n} < 1/2\}) ∪ (Ω ∩ \{x_{n} > L\}) \end{matrix}  (8.21)

if σ is sufficiently small. Define

F̃(x′, x_{n}) = ∫_{0}^{x_{n}} g̃(x′, s)ds  for (x′, x_{n}) ∈ Λ × (−1, ∞).  (8.22)

Then, from (8.21) and (8.22), we have

F̃, F̃_{x_{n}} ∈ C^{α}(\overline{Λ} × [−1, ∞)),  (8.23)
∥F̃∥_{0,α,Λ×(−1,∞)} ≤ Cσ, ∥F̃_{x_{n}}∥_{0,α,Λ×(−1,∞)} ≤ Cσ,  (8.24)
F̃ ≡ 0  in Λ × (−1, 1/2),  (8.25)
F̃ is independent of x_{n} ∈ Λ × (L, ∞).  (8.26)

Note that, from (8.14), for sufficiently small σ, the free boundary S̃ lies within the domain Λ × (−1, 1/2). Then, by (8.25), the function F̃ vanishes on S̃. Using also (8.14) and (8.23), we can rewrite (8.18) as

∫_{C^{+}}(Ã(x, Dũ) + F̃(x)e_{n}) · Dη dx = 0  for any η ∈ C_{0}^{1}(R^{n}).  (8.27)

Thus, ũ is a weak solution of the following conormal problem:

div (Ã(x, Du) + F̃e_{n}) = 0  in C^{+},  (8.28)
(Ã(x, Du) + F̃e_{n}) · ν = 0  on S̃,  (8.29)
(Ã(x, Du) + F̃e_{n}) · ν = 0  on ∂_{l}C^{+}.  (8.30)

Now we make the partial hodograph transform. Define a mapping F : \overline{C^{+}} → R^{n} by

(x′, x_{n}) → (y′, y_{n}) = (x′, ũ(x′, x_{n})).

The nondegeneracy property (8.8) of ũ implies that the mapping F is one-to-one on \overline{C^{+}} and, from (8.8), (8.13), and (8.14),

F(C^{+}) = Λ × (0, ∞), F(S̃) = Λ × {0},

i.e., the free boundary S̃ is mapped to the fixed boundary Λ × {0}. Also, by (8.8) for ũ, there exists a function v ∈ C^{1,α}(\overline{Λ × (0, ∞)}) such that, for (x′, x_{n}) ∈ \overline{C^{+}} and y_{n} ≥ 0,

ũ(x′, x_{n}) = y_{n}  if and only if  v(x′, y_{n}) = x_{n}.  (8.31)

Thus,

F^{−1}(y′, y_{n}) = (y′, v(y′, y_{n})).

Differentiating the identity ũ(x′, v(x′, y_{n})) = y_{n}, which holds for any (x′, y_{n}) ∈ \overline{Λ × (0, ∞)}, we find

v_{y_{n}} > 0 in Λ × (0, ∞)

and

D_{x′}u = −\frac{1}{v_{y_{n}}} D_{y′}v,  u_{x_{n}} = \frac{1}{v_{y_{n}}} ,  (8.32)

where the left-hand and right-hand sides are taken at the points (x′, x_{n}) and F(x′, x_{n}), respectively. In particular, property (8.8) of ũ implies

0 < \frac{1}{2(q_{0}^{−} − q_{0}^{+})} ≤ v_{y_{n}}(y) ≤ \frac{2}{q_{0}^{−} − q_{0}^{+}}  for any y ∈ C^{+}.  (8.33)

From this and (8.15), we get

∥v − v_{0}∥_{1,α,Λ×(0,∞)} ≤ Cσ,  (8.34)

where v_{0}(y) = y_{n}/(q_{0}^{−} − q^{+}). From (8.16) and (8.33),

∥v − v_{0}∥_{C^{1}(Λ×(R,∞))} → 0  as R → ∞.  (8.35)

Now, since ũ(x) is a solution of the conormal problem (8.27) in C^{+}, then v(y) is a solution of the corresponding problem in Λ × (0, ∞). In order to show that this problem has also a conormal structure, we make the change of variables x → y = F(x) in the weak form (8.27) of problem (8.28)–(8.29). In order to do that, we especially need to change the variables in the test function η. For that, we note that the function ψ(y) := η ∘ F^{−1}(y) = η(y′, v(y′, y_{n})) satisfies ψ ∈ C^{1}(\overline{Λ × (0, ∞)}) and, if η ≡ 0 on R^{n} \ B_{R}, then ψ ≡ 0 on (Λ × (0, ∞)) \ B_{R_{1}} for some R_{1}, i.e., ψ = ψ̃|\overline{Λ×(0,∞)} for some ψ̃ ∈ C_{0}^{1}(R^{n}). Similarly, for any ψ ∈ C_{0}^{1}(R^{n}), there exists η ∈ C_{0}^{1}(R^{n}) such that ψ|\overline{R_{+}^{n}} = η ∘ F^{−1} and η(x) = ψ ∘ F(x) for x ∈ \overline{C^{+}}. We differentiate the identity η(x) = ψ(x′, u(x′, x_{n})) and use (8.32) to obtain

D_{x′}η = D_{y′}ψ − \frac{ψ_{y_{n}}}{v_{y_{n}}} D_{y′}v,  η_{y_{n}} = \frac{ψ_{y_{n}}}{v_{y_{n}}} .  (8.36)

Now, in (8.27), we make the change of variables x → y = F(x), use (8.32) and (8.36), note that the Jacobian of F^{−1} is JF^{−1}(y) = v_{y_{n}}(y), and write Ã(x′, x_{n}, p′, p_{n}) for Ã(x, P) and F̃(x′, x_{n}) for F̃(x) to obtain

∫_{Λ×(0,∞)} (B(y′, v, Dv) + F̃(y′, v)e_{n}) · Dψdy = 0  for any ψ ∈ C_{0}^{1}(R^{n}),  (8.37)

where, for y′ ∈ Λ, z ∈ R, P = (p′, p_{n}) ∈ R^{n−1} × R_{+},

\begin{matrix}B^{i}(y′, z, P) = Ã^{i}(y′, z, −\frac{p′}{p_{n}} , \frac{1}{p_{n}})p_{n}  for i = 1, …, n − 1, \\ B^{n}(y′, z, P) = Ã^{n}(y′, z, −\frac{p′}{p_{n}} , \frac{1}{p_{n}}) − ∑_{1}^{n−1} Ã^{i}(y′, z, −\frac{p′}{p_{n}} , \frac{1}{p_{n}})p_{i}. \end{matrix}  (8.38)

Thus, v(y) is a weak solution of the corresponding conormal problem in Λ × (0, ∞).

Note that the conormal problem (8.37) is elliptic for the functions satisfying (8.34) with small σ, that is, there exists λ̃ > 0 such that

λ̃|ξ|^{2} ≤ ∑_{i,j=1}^{n} B_{p_{j}}^{i}(y′, v(y), Dv(y))ξ_{i}ξ_{j} ≤ λ̃^{−1}|ξ|^{2}  (8.39)

for any y ∈ Λ × (0, ∞), ξ ∈ R^{n}, and v satisfying (8.34). To see this, from (8.38), we compute

∑_{i,j=1}^{n} B_{p_{j}}^{i}(y′, z, p)ξ_{i}ξ_{j} = ∑_{i,j=1}^{n} Ã_{p_{j}}^{i}(y′, z, −\frac{p′}{p_{n}} , \frac{1}{p_{n}})ζ_{i}ζ_{j},

where p = (p_{1}, …, p_{n}) ≡ (p′, p_{n}), ζ_{i} = ξ_{i} − p_{i}ξ_{n}/p_{n} for i = 1, …, n − 1, and ζ_{n} = ξ_{n}/p_{n}. Since (8.28) is a uniformly elliptic equation for u = u_{0} := (q_{0}^{−} −q_{0}^{+})x_{n}, it follows from (2.16) and (8.19) that (8.39) holds with the constants depending only on the data for any v satisfying (8.34).

Now, using also the C^{1,α} dependence of B(y, p) on y which follows from φ^{−} ∈ C^{2,α}(\overline{Λ}×[0, ∞)) and Ψ ∈ C^{2,α}(R^{n}), the standard estimates for elliptic equations imply the interior regularity v ∈ C^{2,α}(Λ × (0, ∞)) with

|D^{2}v(y)| ≤ Cσd_{y}^{α−1}  for d_{y} = dist(y, ∂(Λ × R_{+})).  (8.40)

Note that the functions B(y′, z, P) and F̃(y′, z) and the domain Λ × (0, ∞) in (8.37) are independent of a solution φ of Problem (TN).

Conversely, let v(y) is a solution of (8.37) satisfying (8.34) and (8.35) with sufficiently small σ depending only on the data so that (8.33) holds. Then the corresponding solution φ of Problem (TN), satisfying (2.24), can be determined. Indeed, from (8.33),

C^{+} := {(x′, v(x′, y_{n})) : x′ ∈ Λ, y_{n} > 0}  (8.41)

has the form (8.14), and a function ũ ∈ C^{1,α}(\overline{C^{+}}) can be defined such that (8.31) holds. Then (8.27) holds. Moreover, (8.34) implies (8.15). Next, we conclude that Ω^{+} := Ψ(C^{+}) has the form (2.23) and (2.30), and u(x) := ũ(Φ(x)) satisfies (8.12). Moreover, (8.15) implies (8.7). If σ is sufficiently small, then (8.7) and (8.15) imply (8.8) for u and ũ, respectively. Now, (8.35) implies (8.16) and further (8.9). Then it follows that φ, defined by φ = φ^{−} − u in Ω^{+} and φ = φ^{−} in Ω \ Ω^{+}, is a weak solution of Problem (TN) satisfying (2.24).

Thus, the uniqueness of solutions of the conormal problem (8.37) satisfying (8.34) and (8.35) implies the uniqueness of solutions of Problem (TN) satisfying (2.24).

8.3. Uniqueness of Weak Solutions of the Conormal Problem (8.37) Satisfying (8.34) and (8.35). Suppose v_{1} and v_{2} are two solutions of (8.37) in Λ × (0, ∞) satisfying (8.34) and (8.35). We assume that σ is small so that (8.33) holds for v_{1} and v_{2}. Consider the function w = v_{2} − v_{1}. It satisfies

∥w∥_{1,α,Λ×(0,∞)} ≤ 2Cσ  (8.42)

and

∥w∥_{1,0,Λ×(R,∞)} → 0  as R → ∞.  (8.43)

Also, subtracting equations (8.37) for v_{1} and v_{2}, we get

∫_{Λ×(0,∞)} ( ∑_{i,j=1}^{n} a_{ij}(y)w_{y_{j}}φ_{y_{i}} + ∑_{i=1}^{n} b_{i}(y)wφ_{y_{i}} ) dy = 0  (8.44)

for any φ ∈ C_{c}^{1}(R^{n}), where a_{ij} and b_{i} are defined by

\begin{matrix}a_{ij}(y) = ∫_{0}^{1} B_{p_{j}}^{i}(y′, (1 − t)v_{1}(y) + tv_{2}(y), (1 − t)Dv_{1}(y) + tDv_{2}(y))dt, \\ b_{i}(y) = ∫_{0}^{1} (B_{z}^{i}(y′, (1 − t)v_{1}(y) + tv_{2}(y), (1 − t)Dv_{1}(y) + tDv_{2}(y)) \\ +δ_{i}^{n}∂_{n}F̃(y′, (1 − t)v_{1}(y) + tv_{2}(y)))dt. \end{matrix}  (8.45)

Note that a_{ij}, b_{i} ∈ C^{α}(\overline{Λ} × [0, ∞)) with

|a_{ij}(y)| + |b_{i}(y)| ≤ C  for any y ∈ Λ × (0, ∞), i, j = 1, …, n,  (8.46)

which follows from (8.2), (8.11), (8.19), (8.24), (8.33), (8.34), and (8.38). Moreover, using (8.3), (8.20), and (8.26), we see that

b^{i} ≡ 0  on {y_{n} > L_{1}}  for i = 1, …, n,  (8.47)

where L_{1} = 2L/(q_{0}^{−} − q_{0}^{+}). Also, using (8.40) for v_{1} and v_{2} yields

|D^{2}v(y)| ≤ Cσd_{y}^{α−1},  where d_{y} = dist(y, ∂(Λ × (0, ∞))).  (8.48)

Note that, by (8.39), the coefficients a_{ij}(y) satisfy the ellipticity condition (3.35) with constant λ̃ for any y ∈ Λ × (0, ∞).

We first prove that w ≥ 0 by choosing the following test function φ in (8.44): For fixed ε > 0 and R > 0,

φ = \frac{(w − m_{R})^{+}}{(w − m_{R})^{+} + ε},

where

m_{R} := max (0, sup_{y∈Λ×(R,∞)} w(y)) < ∞

by (8.42). Then φ ∈ C^{0,1}(\overline{Λ} × [0, ∞) and φ ≡ 0 on Λ × [R, ∞). Thus we can substitute this φ into (8.44) to obtain

∫_{Λ×(0,R)} (\frac{εa_{ij}w_{y_{i}}^{+}((w − m_{R})^{+})_{y_{j}}}{((w − m_{R})^{+} + ε)^{2}} + \frac{εb_{j}w^{+}((w − m_{R})^{+})_{y_{j}}}{((w − m_{R})^{+} + ε)^{2}} ) dy ≤ 0.

Then using the ellipticity, (8.47), and (8.46) yields

λ ∫_{Λ×(0,R)} |Dlog (1 + (w − m_{R})^{+}/ε)|^{2} dy = λ ∫_{Λ×(0,R)} \frac{|D(w − m_{R})^{+}|^{2}}{((w − m_{R})^{+} + ε)^{2}} dy

≤ ∫_{Λ×(0,R)} (\frac{b_{j}(w − m_{R})^{+}((w − m_{R})^{+})_{y_{j}}}{((w − m_{R})^{+} + ε)^{2}}

+m_{R}b_{j} (\frac{(w − m_{R})^{+}}{(w − m_{R})^{+} + ε})_{y_{j}}) dy

≤ C ∫_{Λ×(0,L_{1})} |Dlog (1 + (w − m_{R})^{+}/ε)| dy
+m_{R} ∫_{∂[Λ×(0,L_{1})]} |b|dS + m_{R} ∫_{Λ×(0,L_{1})} |Db|dy

≤ C (∫_{Λ×(0,L_{1})} |Dlog (1 + (w − m_{R})^{+}/ε)|^{2} dy)^{1/2} + C,

where we used (8.45) and (8.48) in the last estimate. Thus, for R > L_{1},

∫_{Λ×(0,R)} |Dlog (1 + (w − m_{R})^{+}/ε)|^{2} dy ≤ C.

Since log (1 + (w − m_{R})^{+}/ε) = 0 on Λ × {y_{n} = R}, it follows from the Poincaré inequality that

∫_{Λ×(0,R)} |log (1 + (w − m_{R})^{+}/ε)|^{2} dy ≤ C,

where C depends on the data and R, but is independent of ε. Letting ε → 0, we conclude (w − m_{R})^{+} ≡ 0 in Λ × (0, R), which, from the definition of m_{R}, implies w ≤ m_{R} in Λ × (0, ∞). Letting R → ∞ and noting that lim_{R→∞} m_{R} = 0 by (8.43), we get w ≤ 0 in Λ × (0, ∞). Similarly,

w ≥ 0 in Λ × (0, ∞). Thus the uniqueness is proved.

8.4. Stability. As a consequence of the uniqueness, nondegeneracy, and regularity of solutions of the free boundary problem, we have the following stability theorem.

Theorem 8.1. There exist a constant σ_{0} > 0 and a nonnegative nondecreasing function G ∈ C([0, ∞)) with G(0) = 0, depending only on n, α, γ, C_{0}, Λ, L, and q_{0}^{+}, such that, if

(i) 0 < κ < σ < σ_{0}; (ii) Ψ : R^{n} → R^{n} satisfies (2.16), and (2.17), and Ψ̂ : R^{n} → R^{n} satisfies (2.17) and

∥Ψ − Ψ̂∥_{3,α,Λ×(− \frac{1}{2} , \frac{1}{2} )} ≤ κ;  (8.49)

(iii) φ^{−}(x) satisfies (2.32) and φ̂^{−}(x) satisfies

∥φ^{−} ∘ Ψ^{−1} − φ̂^{−} ∘ Ψ̂^{−1}∥_{2,α,Λ×(− \frac{1}{2} , \frac{1}{2} )} ≤ κ,  (8.50)

then the unique solutions φ(x) and φ̂(x) of Problem (TN) for the nozzles defined by the mappings Ψ and Ψ̂ in (2.14) and supersonic upstream flows φ^{−}(x) and φ̂^{−}(x), respectively, satisfy

∥f̃_{φ} − f̃φ̂∥_{1,α,Λ} ≤ G(κ),  (8.51)

where f̃_{φ}(x′) and f̃φ̂(x′) are the free boundary functions of the cylindric domains C^{+}(φ) := Φ(Ω^{+}(φ)) and C^{+}(φ̂(x)) := Φ̂(Ω^{+}(φ̂)) in (8.14), respectively, where Φ = Ψ^{−1} and Φ̂ = Ψ̂^{−1}.

Its proof repeats the one of [4, Theorem 6.1]. See also the remarks on the proof of the stability part of [5, Theorem 2.3]. In the present case, as an additional step, the mapping from Ω to C is involved.

Appendix A. Existence of Smooth Supersonic Upstream Flows in the Nozzle

In this paper, we require a local C^{1,α} supersonic solution φ^{−} in Sections 3–7 and C^{2,α} supersonic solution φ^{−} in Section 8 in the domain Ω_{1} := Ω ∩ {−1 < x_{n} ≤ 1} of the initial-boundary value problem (2.18)–(2.19) for the nonlinear wave equation (1.1) such that

∥φ^{−} − φ_{0}^{−}∥_{k,α,Ω_{1}} ≤ C_{0}σ,  k = 1 or 2,  (A.1)

for some C_{0} > 0 independent of σ, when σ is sufficiently small. This is a direct corollary of the standard local theory of smooth solutions of the initial-boundary value problem for the nonlinear wave equations.

Proposition A.1. Let the nozzle boundary satisfy (2.15) and (2.16). Let the Cauchy data (φ_{e}^{−}, ψ_{e}^{−}) on ∂_{o}Ω satisfies

∥φ_{e}^{−} − q_{0}^{−}x_{n}∥_{H^{s+k}} + ∥ψ_{e}^{−} − q_{0}^{−}∥_{H^{s+k−1}} ≤ σ,  k = 1 or 2,  (A.2)

for some integer s > n/2 + 1 and the compatibility conditions on the intersection \overline{∂_{l}Ω} ∩ \overline{∂_{o}Ω} up to the (s + k)^{th}–order. Then there exist σ_{0} > 0 and C_{0} > 0 such that there exists a unique classical solution φ^{−} ∈ C^{k+1}(\overline{Ω_{1}}) of problem (2.18)–(2.19) for the nonlinear wave equation (1.1), respectively for k = 1 or 2, such that, when σ ≤ σ_{0},

∥φ^{−} − φ_{0}^{−}∥_{k+1,0,Ω_{1}} ≤ C_{0}σ.  (A.3)

If, in addition, the conditions (2.15), (2.16), and (A.2), as well as the compatibility conditions holds for higher order derivatives, then the solution φ^{−} has correspondingly higher regularity.

As in Section 8, we set

w(x) := φ^{−}(x) − q_{0}^{−}x_{n}.

Then w satisfies the following initial-boundary value problem:

divA(Dw) = 0,  (A.4)
(w, w_{x_{n}})|_{x_{n}=−1} = (φ_{e}^{−} + q_{0}^{−}, ψ_{e}^{−} − q_{0}^{−}),  (A.5)
w_{ν}|_{∂_{l}Ω} = −q_{0}^{−}ν_{n},  (A.6)

as well as the compatibility conditions up to the (s + k)^{th}–order, where

A(P) = ρ(|P + q_{0}^{−}e_{n}|^{2})(P + q_{0}^{−}e_{n}) − ρ((q_{0}^{−})^{2})q_{0}^{−}e_{n}.

Since q_{0}^{−} > c_{∗}, then (A.4) is strictly hyperbolic, nonlinear wave equations with respect to the x_{n}-direction (whose role is as the time direction) for a small perturbation of the trivial solution u = 0. By the standard energy estimates and the iteration procedure (cf. [28, 29, 31] and the references cited therein), then, for small σ > 0, there exists a unique solution

w(x′, x_{n}) ∈ ∩_{j=0}^{s+k}C^{j}([−1, 1]; H^{s+1−j}(Ω̄))

such that

∑_{j=0}^{s+k} ∥w∥_{C^{j}([−1,1];H^{s+1−j}(Ω̄))} ≤ Cσ,

for k = 1 or 2, respectively. Since s > n/2 + 1, the Sobolev embedding theorem implies (A.3).

Acknowledgments. Gui-Qiang Chen’s research was supported in part by the National Science Foundation under Grants DMS-0244473, DMS-0204225, DMS-0204455, and INT-9987378 and an Alexandre von Humboldt Foundation Fellowship. Mikhail Feldman’s research was supported in part by the National Science Foundation under Grants DMS-0200644 and DMS-0074037.

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Department of Mathematics, Northwestern University, Evanston, IL 60208-2730

   E-mail address: gqchen@math.northwestern.edu

Department of Mathematics, University of Wisconsin, Madison, WI 53706

   E-mail address: feldman@math.wisc.edu