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: