Improvement of flatness for vector valued free boundary problems

Daniela De Silva; Giorgio Tortone; Daniela De Silva; Giorgio Tortone

doi:10.3934/mine.2020027

Mathematics in Engineering

2020, Volume 2, Issue 4: 598-613. doi: 10.3934/mine.2020027

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Improvement of flatness for vector valued free boundary problems

Daniela De Silva ^{1
,
,},
Giorgio Tortone ^{2
,
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1.
Department of Mathematics, Barnard College, Columbia University, New York, NY 10027, USA
2.
Dipartimento di Matematica, Alma Mater Studiorum Universita di Bologna, Piazzà di Porta San Donato 5, 40126 Bologna, Italy

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 C^{1, 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].
- one-phase free boundary problem,
- Harnack inequality,
- vectorial problem,
- viscosity solution,
- improvement of flatness
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

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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, 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].

References

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