# Geometry of quasiminimal phase transitions

created by farina on 24 Nov 2007

[BibTeX]

Accepted Paper

Inserted: 24 nov 2007

Journal: Calc. Var. Partial Differential Equations
Year: 2007

Abstract:

We consider the quasiminima of the energy functional

$$\int\Omega A(x,\nabla u)+F(x,u)\,dx\,,$$ where $A(x,\nabla u)\sim \nabla u ^p$ 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.

{F}rom 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.