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 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.