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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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    [1] Caffarelli LA, Alt HW (1981) Existence and regularity for a minimum problem with free boundary. J Reine Angew Math 325: 105-144.
    [2] Caffarelli LA (1987) A Harnack inequality approach to the regularity of free boundaries. Part I. Lipschitz free boundaries are C1,α. Rev Mat Iberoamericana 3: 139-162.
    [3] Caffarelli LA (1988) A Harnack inequality approach to the regularity of free boundaries. Part III. Existence theory, compactness, and dependence on x. Ann Scuola Norm Sci 15: 583-602.
    [4] Caffarelli LA (1989) A Harnack inequality approach to the regularity of free boundaries. Part II. Flat free boundaries are Lipschitz. Commun Pure Appl Math 42: 55-78.
    [5] Caffarelli LA, Roquejoffre JM, Sire Y (2010) Variational problems for free boundaries for the fractional Laplacian. J Eur Math Soc 12: 1151-1179.
    [6] Caffarelli LA, Shahgholian H, Yeressian K (2018) A minimization problem with free boundary related to a cooperative system. Duke Math J 167: 1825-1882. doi: 10.1215/00127094-2018-0007
    [7] De Philippis G, Spolaor L, Velichkov B (2019) Regularity of the free boundary for the two-phase Bernoulli problem. arXiv:1911.02165.
    [8] De Silva D (2011) Free boundary regularity for a problem with right hand side. Interface Free Bound 13: 223-238.
    [9] De Silva D, Ferrari F, Salsa S (2014) On two phase free boundary problems governed by elliptic equations with distributed sources. Discrete Contin Dyn Syst Ser S 7: 673-693.
    [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
    [13] Kriventsov D, Lin FH (2019) Regularity for shape optimizers: The degenerate case. Commun Pure Appl Math 72: 1678-1721. doi: 10.1002/cpa.21810
    [14] Mazzoleni D, Terracini S, Velichkov B (2017) Regularity of the optimal sets for some spectral functionals. Geom Funct Anal 27: 373-426. doi: 10.1007/s00039-017-0402-2
    [15] Mazzoleni D, Terracini S, Velichkov B (2020) Regularity of the free boundary for the vectorial Bernoulli problem. Anal PDE 13: 741-763. doi: 10.2140/apde.2020.13.741
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