Partial regularity for a minimum problem with free boundary

Abstract

Regularity of the free boundary ∂{u > 0} of a non-negative minimum u of the functional$\upsilon \mapsto \int\limits_\Omega {\left( {\left| {\nabla \upsilon } \right|^2 + Q^2 \chi _{\left\{ {\upsilon > 0} \right\}} } \right)}$, where Ω is an open set in ℝⁿ and Q is a strictly positive Hölder-continuous function, is still an open problem for n ≥ 3.

By means of a new monotonicity formula we prove that the existence of singularities is equivalent to the existence of an absolute minimum u* such that the graph of u* is a cone with vertex at 0, the free boundary ∂{u* > 0} has one and only one singularity, and the set {u* > 0} minimizes the perimeter among all its subsets.

This leads to the following partial regularity: there is a maximal dimension k* ≥ 3 such that for n < k* the free boundary ∂{u > 0} is locally in Ω a C^1,α-surface, for n = k* the singular set Σ:= ∂{u > 0} − ∂_red{u > 0} consists at most of in Ω isolated points, and for n > k* the Hausdorff dimension of the singular set Σ is less than n - k*.