Improvement of flatness for vector valued free boundary problems

TL;DR

This paper proves that flatness of the free boundary in vector-valued Bernoulli problems guarantees its $C^{1,eta}$ regularity, extending known scalar results to the vector case using a viscosity approach.

Contribution

It introduces a direct viscosity method for vectorial free boundary problems, avoiding reduction to scalar problems and enabling future extensions to fractional Laplacian cases.

Findings

Flat free boundary implies $C^{1,eta}$ regularity in vector problems.

The viscosity approach directly addresses vectorial problems without scalar reduction.

Potential application to fractional Laplacian free boundary problems.

Abstract

For a vectorial Bernoulli-type free boundary problem, with no sign assumption on the components, we prove that flatness of the free boundary implies $C^{1,\alpha}$ regularity, as well-known in the scalar case \cite{AC,C2}. While in \cite{MTV2} the same result is obtained for minimizing solutions by using a reduction to the scalar problem, and the NTA structure of the regular part of the free boundary, our result uses directly a viscosity approach on the vectorial problem, in the spirit of \cite{D}. We plan to use the approach developed here in vectorial free boundary problems involving a fractional Laplacian, as those treated in the scalar case in \cite{DR, DSS}.