79 research outputs found
Free boundary regularity for a multiphase shape optimization problem
In this paper we prove a regularity result in dimension two
for almost-minimizers of the constrained one-phase Alt-Caffarelli and the
two-phase Alt-Caffarelli-Friedman functionals for an energy with variable
coefficients. As a consequence, we deduce the complete regularity of solutions
of a multiphase shape optimization problem for the first eigenvalue of the
Dirichlet-Laplacian up to the fixed boundary. One of the main ingredient is a
new application of the epiperimetric-inequality of Spolaor-Velichkov [CPAM,
2018] up to the boundary. While the framework that leads to this application is
valid in every dimension, the epiperimetric inequality is known only in
dimension two, thus the restriction on the dimension
Uniqueness of the blow-up at isolated singularities for the Alt-Caffarelli functional
In this paper we prove uniqueness of blow-ups and -regularity for
the free-boundary of minimizers of the Alt-Caffarelli functional at points
where one blow-up has an isolated singularity. We do this by establishing a
(log-)epiperimetric inequality for the Weiss energy for traces close to that of
a cone with isolated singularity, whose free-boundary is graphical and smooth
over that of the cone in the sphere. With additional assumptions on the cone,
we can prove a classical epiperimetric inequality which can be applied to
deduce a regularity result. We also show that these additional
assumptions are satisfied by the De Silva-Jerison-type cones, which are the
only known examples of minimizing cones with isolated singularity. Our approach
draws a connection between epiperimetric inequalities and the \L ojasiewicz
inequality, and, to our knowledge, provides the first regularity result at
singular points in the one-phase Bernoulli problem.Comment: 37 pages. To appear in Duke Math Journa
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