University of Bath

University of Bristol

Imperial College London

University of Oxford

Engineering and Physical Sciences Research Council

The University of Warwick

Mathematics Taught Course Centre

Elliptic-Hyperbolic Partial Differential Equations

October - December 2013,  Tuesdays 14:00-16:00   Prof. Gui-Qiang G. Chen and Dr. Wei Xiang

Partial differential equations of mixed elliptic-hyperbolic type arise naturally in mechanics, geometry, analysis, mathematical physics, and other areas. The solution of some fundamental issues in these areas greatly requires a deep understanding of nonlinear PDEs of mixed elliptic-hyperbolic type. Important examples include transonic flow equations in fluid mechanics and the Gauss-Codazzi equations for isometric embedding in differential geometry. This course is an introduction to some facets of mathematical techniques/approaches for solving PDEs of mixed elliptic-hyperbolic type and their applications to mechanics, geometry, analysis, and other areas.


The topics include: introduction;  linear degenerate elliptic equations; nonlinear degenerate elliptic equations; fixed point theorems, degree theory and applications; nonlinear conservation laws of mixed type and shock reflection-diffraction problems;  mathematics of shock reflection-diffraction and free boundary problems;  further topics on PDEs of mixed type.


Basic PDE theory/functional analysis or the equivalent is the only essential prerequisite. However, some familiarity with basic nonlinear PDEs, fluid mechanics, and differential geometry is desirable.


Lecture Notes:

Lecture 1
Lecture 2
Lecture 3
Lecture 4


Lecture 5


Lecture 6


Lecture 7-1

Lecture 7-2



Topics & References:  See  Lecture 1