Calculus of Variations and Geometric Measure Theory

D. De Silva - G. Tortone

A vectorial problem with thin free boundary

created by tortone on 01 Nov 2023


Published Paper

Inserted: 1 nov 2023
Last Updated: 1 nov 2023

Journal: Calculus of Variations and Partial Differential Equations
Year: 2023
Doi: 10.1007/s00526-023-02561-z

ArXiv: 2010.05782 PDF


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.