Below is the ascii version of the abstract for 06-267.
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Alberto Farina, Enrico Valdinoci
Geometry of quasiminimal phase transitions
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
We show that the Lipschitz quasiminima,
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
hypersurfaces of quasimimal
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
inequalities for level sets are obtained
and then, if the limit surface if flat,
so are the level sets of the solution.