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.
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