Geometric PDEs in the Grushin Plane: Weighted Inequalities and Flatness of Level Sets

Abstract

A geometric Sobolev-Poincare inequality for stable solutions of semilinear partial differential equations (PDEs) in the Grushin plane will be obtained. Such inequality will bound the weighted L 2 -norm of a test function by a weighted L 2 -norm of its gradient, and the weights will be interesting geometric quantities related to the level sets of the solution. From this, we shall see that a geometric PDE holds on the level sets of stable solutions. We shall study in detail the particular case of local minimizers of a Ginzburg-Landau-Allen-Cahn-type phase transition model and provide for them some one-dimensional symmetry results.