Geometrization Program of Semilinear Elliptic Equations

Understanding the entire solutions of nonlinear elliptic equations in RN such as (0.1) Δu+ f(u) = 0 in R , is 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). 1. Part I: Geometrization Program of Allen-Cahn Equation In this section, we survey the studies on entire solutions of Allen-Cahn equation. 1.1. Background. The Allen-Cahn equation in R is the semilinear elliptic problem (1.1) Δu + u− u = 0 in R . Originally formulated in the description of bi-phase separation in fluids [16] and ordering in binary alloys [2], Equation (1.1) has received extensive mathematical study. It is a prototype for the modeling of phase transition phenomena in a variety of contexts. Introducing a small positive parameter ε and writing v(x) := u(ε−1x), we get the scaled version of (1.1), (1.2) εΔv + v − v = 0 in R . c © 2012 American Mathematical Society and International Press