Research article

Existence of viscosity solutions to two-phase problems for fully nonlinear equations with distributed sources

  • Received: 16 July 2018 Accepted: 22 October 2018 Published: 25 October 2018
  • In this paper we construct a viscosity solution of a two-phase free boundary problem for a class of fully nonlinear equation with distributed sources, via an adaptation of the Perron method. Our results extend those in [Ca arelli, 1988], [Wang, 2003] for the homogeneous case, and of [De Silva, Ferrari, Salsa, 2015] for divergence form operators with right hand side.

    Citation: Sandro Salsa, Francesco Tulone, Gianmaria Verzini. Existence of viscosity solutions to two-phase problems for fully nonlinear equations with distributed sources[J]. Mathematics in Engineering, 2019, 1(1): 147-173. doi: 10.3934/Mine.2018.1.147

    Related Papers:

  • In this paper we construct a viscosity solution of a two-phase free boundary problem for a class of fully nonlinear equation with distributed sources, via an adaptation of the Perron method. Our results extend those in [Ca arelli, 1988], [Wang, 2003] for the homogeneous case, and of [De Silva, Ferrari, Salsa, 2015] for divergence form operators with right hand side.


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    [1] Caffarelli L, Salsa S (2005) A geometric approach to free boundary problems (Volume 68). American Mathematical Soc.
    [2] Caffarelli LA (1987) A Harnack inequality approach to the regularity of free boundaries. Part I: Lipschitz free boundaries are C1,ɑ. Rev Math Iberoam 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. Comm Pure Appl Math 42: 55–78.
    [5] Caffarelli LA Cabré X (1995) Fully nonlinear elliptic equations (Volume 43). American Mathematical Soc.
    [6] Caffarelli LA, Jerison D, Kenig CE (2002) Some new monotonicity theorems with applications to free boundary problems. Ann Math 155: 369–404. doi: 10.2307/3062121
    [7] De Silva D (2011) Free boundary regularity for a problem with right hand side. Interface Free Bound 13: 223–238.
    [8] De Silva D, Ferrari F, Salsa S (2015) Free boundary regularity for fully nonlinear non-homogeneous two-phase problems. J Math Pure Appl 103: 658–694. doi: 10.1016/j.matpur.2014.07.006
    [9] De Silva D, Ferrari F, Salsa S (2015) Perron's solutions for two-phase free boundary problems with distributed sources. Nonlinear Anal 121: 382–402. doi: 10.1016/j.na.2015.02.013
    [10] De Silva D, Ferrari F, Salsa S (2017) Two-phase free boundary problems: from existence to smoothness. Adv Nonlinear Stud 17: 369–385.
    [11] Fabes E, Garofalo N, Marín-Malave S, et al. (1988) Fatou theorems for some nonlinear elliptic equations. Rev Math Iberoam 4: 227–251.
    [12] Gilbarg D, Trudinger NS (1983) Elliptic partial differential equations of second order. second edition, Springer-Verlag, Berlin.
    [13] Lee KA (1998) Obstacle problems for the fully nonlinear elliptic operators. PhD thesis, Courant Institute, New York University.
    [14] Matevosyan N, Petrosyan A (2011) Almost monotonicity formulas for elliptic and parabolic operators with variable coefficients. Comm Pure Appl Math 64: 271–311. doi: 10.1002/cpa.20349
    [15] Silva DD, Savin O (2017) Lipschitz regularity of solutions to two-phase free boundary problems. Int Math Res Notices.
    [16] Silvestre L, Sirakov B (2014) Boundary regularity for viscosity solutions of fully nonlinear elliptic equations. Commun Part Diff Equ 39: 1694–1717. doi: 10.1080/03605302.2013.842249
    [17] Wang PY (2000) Regularity of free boundaries of two-phase problems for fully nonlinear elliptic equations of second order. I. Lipschitz free boundaries are C1,ɑ. Comm Pure Appl Math 53: 799– 810.
    [18] Wang PY (2002) Regularity of free boundaries of two-phase problems for fully nonlinear elliptic equations of second order. II. Flat free boundaries are Lipschitz. Commun Part Diff Eq 27: 1497– 1514.
    [19] Wang PY (2003) Existence of solutions of two-phase free boundary problems for fully nonlinear elliptic equations of second order. J Geom Anal 13: 715–738.
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