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Improvement of flatness for vector valued free boundary problems

Daniela De Silva Giorgio Tortone

*Corresponding author: Daniela De Silva, Giorgio Tortone desilva@math.columbia.edu;giorgio.tortone@unibo.it

MinE2020,4,598doi:10.3934/mine.2020027

For a vectorial Bernoulli-type free boundary problem, with no sign assumption on the components, we prove that flatness of the free boundary implies C1,α regularity, as well-known in the scalar case [1, 4]. While in [15] 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 [8]. 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 [10, 11].

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