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.