Geometrization program of semilinear elliptic equations

Abstract

Abstract. Understanding the entire solutions of nonlinear elliptic equations in R N such asis a basic problem in PDE research. This is the context of various classical results in literature like the Gidas-Ni-Nirenberg theorems on radial symmetry, Liouville type theorems, or the achievements around De Giorgi's conjecture.In those results, the geometry of level sets of the solutions turns out to be a posteriori very simple (planes or spheres). On the other hand, problems of the form (0.1) do have solutions with more interesting patterns, and the structure of their solution sets has remained mostly a mystery. A major aspect of our research program is to bring ideas from Differential Geometry into the analysis and construction of entire solutions for two important equations: (1) the AllenCahn equation and (2) the nonlinear Schrodinger equation (NLS). Though simple-looking, they are typical representatives of two classes of semilinear elliptic problems. The structure of entire solutions is quite rich. In this survey, we shall establish an intricate correspondence between the study of entire solutions of some scalar equations and the theories of minimal surfaces and constant mean curvature surfaces (CMC).