Free boundary regularity for a multiphase shape optimization problem

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.