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.

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.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11585/78143
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