Regularity of the free boundary for the two-phase Bernoulli problem

TL;DR

This paper establishes regularity results for the free boundary in the two-phase Bernoulli problem, extending classical analysis and also applying findings to multiphase spectral optimization of the Dirichlet Laplacian.

Contribution

It provides a comprehensive regularity theorem for the free boundary in the two-phase Bernoulli problem, completing prior foundational work and applying results to spectral optimization.

Findings

Proved regularity of the free boundary for the two-phase Bernoulli problem.

Showed regularity of minimizers in multiphase spectral optimization.

Extended classical free boundary analysis to new multiphase contexts.

Abstract

We prove a regularity theorem for the free boundary of minimizers of the two-phase Bernoulli problem, completing the analysis started by Alt, Caffarelli and Friedman in the 80s. As a consequence, we also show regularity of minimizers of the multiphase spectral optimization problem for the principal eigenvalue of the Dirichlet Laplacian.