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.
F. Ferrari, E. Valdinoci (2009). Geometric PDEs in the Grushin Plane: Weighted Inequalities and Flatness of Level Sets. INTERNATIONAL MATHEMATICS RESEARCH NOTICES, 22, 4232-4270 [10.1093/imrn/rnp088].
Geometric PDEs in the Grushin Plane: Weighted Inequalities and Flatness of Level Sets
FERRARI, FAUSTO;
2009
Abstract
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
