Alberto Farina, Enrico Valdinoci
Geometry of quasiminimal phase transitions
(278K, PostScript)

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. From this, it follows that
the level sets
of the rescaled quasiminima approach
locally uniformly
hypersurfaces of quasimimal
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.