Minimization of a fractional perimeter-Dirichlet integral functional

TL;DR

This paper studies a minimization problem combining Dirichlet energy with a nonlocal perimeter, establishing regularity of minimizers and free boundaries through blow-up analysis and related techniques.

Contribution

It provides new regularity results and analytical tools for minimizers of a fractional perimeter-Dirichlet functional, advancing understanding of nonlocal free boundary problems.

Findings

Regularity results for minimizers

Regularity of free boundaries

Density estimates and monotonicity formulas

Abstract

We consider a minimization problem that combines the Dirichlet energy with the nonlocal perimeter of a level set, namely $$ \int_\Om |\nabla u(x)|^2\,dx+\Per\Big(\{u > 0\},\Om \Big),$$ with $\sigma\in(0,1)$. We obtain regularity results for the minimizers and for their free boundaries $\p \{u>0\}$ using blow-up analysis. We will also give related results about density estimates, monotonicity formulas, Euler-Lagrange equations and extension problems.