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 boundary of a fixed domain that acts as a geometric inclusion constraint. One of the main ingredients is a new application of the (one-phase) epiperimetric inequality up to the boundary of the constraint. 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.