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International Mathematics Research Notices Advance Access originally published online on June 23, 2009
International Mathematics Research Notices (2009) 2009:4232-4270, doi:10.1093/imrn/rnp088 published on October 27, 2009
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© The Author 2009. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org

Geometric PDEs in the Grushin Plane: Weighted Inequalities and Flatness of Level Sets

Fausto Ferrari1 and Enrico Valdinoci2

1 Dipartimento di Matematica dell'Università Piazza di Porta S. Donato, 5, 40126 Bologna, Italy and C.I.R.A.M. Via Saragozza, 8, 40123 Bologna, Italy
2 Dipartimento di Matematica, dell'Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy

Correspondence: Correspondence to be sent to: valdinoci{at}mat.uniroma2.it

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 L2-norm of a test function by a weighted L2-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.

Communicated by Prof. Carlos Kenig


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