Calculus of Variations and Geometric Measure Theory

D. De Silva - G. Tortone

Improvement of flatness for vector valued free boundary problems

created by tortone on 01 Nov 2023


Published Paper

Inserted: 1 nov 2023
Last Updated: 1 nov 2023

Journal: Mathematics in Engineering
Year: 2020
Doi: 10.3934/mine.2020027

ArXiv: 1909.01290 PDF


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}.