### Mathematics in Engineering

2020, Issue 4: 598-613. doi: 10.3934/mine.2020027
Research article Special Issues

# Improvement of flatness for vector valued free boundary problems

• Received: 06 September 2019 Accepted: 26 February 2020 Published: 19 May 2020
• 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].

Citation: Daniela De Silva, Giorgio Tortone. Improvement of flatness for vector valued free boundary problems[J]. Mathematics in Engineering, 2020, 2(4): 598-613. doi: 10.3934/mine.2020027

### Related Papers:

• 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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