A vectorial problem with thin free boundary

TL;DR

This paper investigates a vectorial version of the thin free boundary problem, analyzing regularity, structure, and properties of minimizers and the free boundary, including a decomposition into regular and singular parts.

Contribution

It introduces a vectorial formulation of the thin free boundary problem and establishes regularity results, free boundary structure, and a new approach for regularity using viscosity solutions.

Findings

Free boundary decomposes into regular and singular parts.

Regular part of the free boundary is proven to be $C^{1,eta}$.

Optimal regularity and nondegeneracy of minimizers are established.

Abstract

We consider the vectorial analogue of the thin free boundary problem introduced in \cite{CRS} as a realization of a nonlocal version of the classical Bernoulli problem. We study optimal regularity, nondegeneracy, and density properties of local minimizers. Via a blow-up analysis based on a Weiss type monotonicity formula, we show that the free boundary is the union of a "regular" and a "singular" part. Finally we use a viscosity approach to prove $C^{1,\alpha}$ regularity of the regular part of the free boundary.