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 language | English |
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Pages (from-to) | 4232-4270 |
Number of pages | 39 |
Journal | International Mathematics Research Notices |
Volume | 2009 |
Issue number | 22 |
DOIs | |
Publication status | Published - 1 Dec 2009 |
Externally published | Yes |