Nonlinear Partial Differential Equations of Mixed Type: Analysis and Connections

Three of the basic types of linear partial differential equations (PDEs) are elliptic, hyperbolic, and parabolic, following the standard classification for linear PDEs. Linear theories of PDEs of these types have been considerably better developed. On the other hand, many nonlinear PDEs arising in Mathematics and Science naturally are of mixed type. The solution of some longstanding fundamental problems greatly requires a deep understanding of such nonlinear PDEs of mixed type, especially mixed elliptic-hyperbolic type. Important examples include shock reflection/diffraction problems in fluid mechanics (the Euler equations) and isometric embedding problems in differential geometry (the Gauss-Codazzi-Ricci equations), among many others. In this talk, we will present some old and new underlying connections of nonlinear PDEs of mixed type with the longstanding fundamental problems and will then discuss some recent developments in the analysis of these nonlinear PDEs through the examples with emphasis on developing/identifying unified approaches, ideas, and techniques for dealing with the mixed-type problems. Some most recent developments, further perspectives, and open problems in this direction will also be addressed.