Geometry of quasiminimal phase transitions

Alberto Farina, Enrico Valdinoci

Research output: Contribution to journalArticlepeer-review

15 Citations (Scopus)

Abstract

We consider the quasiminima of the energy functional ∫ Ω A(x, ∇ u) + F(x, u) dx, where A(x, ∇ u) ∼ |∇ u|p and F is a double-well potential. We show that the Lipschitz quasiminima, which satisfy an equipartition of energy condition, possess density estimates of Caffarelli-Cordoba-type, that is, roughly speaking, the complement of their interfaces occupies a positive density portion of balls of large radii. From this, it follows that the level sets of the rescaled quasiminima approach locally uniformly hypersurfaces of quasiminimal perimeter. If the quasiminimum is also a solution of the associated PDE, the limit hypersurface is shown to have zero mean curvature and a quantitative viscosity bound on the mean curvature of the level sets is given. In such a case, some Harnack-type inequalities for level sets are obtained and then, if the limit surface if flat, so are the level sets of the solution.

Original languageEnglish
Pages (from-to)1-35
Number of pages35
JournalCalculus of Variations and Partial Differential Equations
Volume33
Issue number1
DOIs
Publication statusPublished - 1 Sep 2008
Externally publishedYes

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