Free boundary regularity for a multiphase shape optimization problem

2018/10/16 by Spolaor, Luca, Trey, Baptiste, Velichkov, Bozhidar
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1810.06963

Abstract

In this paper we prove a C^1,α 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.

Related

• Regularity of the free boundary for the two-phase Bernoulli problem
• Regularity of the optimal sets for a class of integral shape functionals
• Regularity for Shape Optimizers: The Nondegenerate Case
• Regularity of the optimal sets for some spectral functionals
• Almost minimizers of the one-phase free boundary problem