Geometric pdes in the grushin plane: Weighted inequalities and flatness of level sets

Fausto Ferrari, Enrico Valdinoci

Research output: Contribution to journalArticle

8 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)4232-4270
Number of pages39
JournalInternational Mathematics Research Notices
Volume2009
Issue number22
DOIs
Publication statusPublished - 1 Dec 2009
Externally publishedYes

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