Nonlinear Partial Differential Equations of Mixed Type: From the Riemann Problem to the Isometric Embedding Problem

Three of the basic types of 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 are not of a single type, but are naturally of mixed type. The solution of several longstanding fundamental problems greatly requires a deep understanding of such nonlinear PDEs of mixed type, especially mixed elliptic-hyperbolic type. Important examples include the multidimensional Riemann problem (formulated by Riemann in 1860 for the one-dimensional case) and related shock reflection/diffraction problems in fluid dynamics (the compressible Euler equations), and the isometric embedding problem 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 from the Riemann problem to the isometric embedding problem 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.