Abstract

As is well-known, two of the basic types of linear partial differential equations (PDEs) are elliptic and hyperbolic, following the classification for linear PDEs proposed by Jacques Hadamard in the 1920s; and linear theories of PDEs of these two types have been relatively well developed. On the other hand, many nonlinear PDEs arising in geometry, mechanics, and other areas naturally are of mixed elliptic-hyperbolic type. The solution of some longstanding fundamental problems in these areas greatly requires a deep understanding of such nonlinear PDEs of mixed type. Important examples include shock reflection-diffraction problems in fluid mechanics (the Euler equations) and isometric embedding problems in differential geometry and materials science (the Gauss-Codazzi-Ricci equations), among many others. In this talk we will present natural connections of nonlinear PDEs of mixed elliptic-hyperbolic type with these longstanding problems and will then discuss some recent developments in the analysis of these nonlinear PDEs through the examples with emphasis on developing and identifying unified approaches, ideas, and techniques for dealing with the mixed-type problems. Further trends, perspectives, and open problems in this direction will also be addressed.