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

A geometric Sobolev–Poincaré inequality for stable solutions of semilinear partial differential equations (PDEs) in the Grushin plane will be obtained. Such inequality will bound the weighted L²-norm of a test function by a weighted L²-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.