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Mathematics in Engineering

2019, Volume 1, Issue 1: 147-173. doi: 10.3934/Mine.2018.1.147
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Research article

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

  • 1.
    Dipartimento di Matematica, Politecnico di Milano, piazza Leonardo da Vinci 32, 20133 Milano,Italy
  • 2.
    Dipartimento di Matematica ed Informatica, Università di Palermo via Archirafi 34, 90123 Palermo, Italy
  • 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

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  • 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.




    [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.
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    [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.
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