Calculus of Variations and Geometric Measure Theory

L. Spolaor - B. Trey - B. Velichkov

Free boundary regularity for a multiphase shape optimization problem

created by spolaor on 16 Oct 2018
modified by velichkov on 04 Nov 2019


Accepted Paper

Inserted: 16 oct 2018
Last Updated: 4 nov 2019

Journal: Communications in Partial Differential Equations
Year: 2018


In this paper we prove a $C^{1,\alpha}$ 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.

Keywords: Regularity of free boundary, shape optimization