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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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    [10] De Silva D, Roquejoffre JM (2012) Regularity in a one-phase free boundary problem for the fractional Laplacian. Ann I H Poincare An 29: 335-367. doi: 10.1016/j.anihpc.2011.11.003
    [11] De Silva D, Savin O, Sire Y (2014) A one-phase problem for the fractional Laplacian: Regularity of flat free boundaries. Bull Inst Math Acad Sin 9: 111-145.
    [12] Kriventsov D, Lin FH (2018) Regularity for shape optimizers: The nondegenerate case. Commun Pure Appl Math 71: 1535-1596. doi: 10.1002/cpa.21743
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