Calculus of Variations and Geometric Measure Theory
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A. Farina - E. Valdinoci

Geometry of quasiminimal phase transitions

created by farina on 24 Nov 2007


Accepted Paper

Inserted: 24 nov 2007

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


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.


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