JournalsaihpcVol. 34, No. 4pp. 899–932

The two membranes problem for different operators

  • L. Caffarelli

    Department of Mathematics, University of Texas at Austin, Austin, TX 78712, USA

  • D. De Silva

    Department of Mathematics, Barnard College, Columbia University, New York, NY 10027, USA

  • O. Savin

    Department of Mathematics, Columbia University, New York, NY 10027, USA
The two membranes problem for different operators cover
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Abstract

We study the two membranes problem for different operators, possibly nonlocal. We prove a general result about the Hölder continuity of the solutions and we develop a viscosity solution approach to this problem. Then we obtain regularity of the solutions provided that the orders of the two operators are different. In the special case when one operator coincides with the fractional Laplacian, we obtain the optimal regularity and a characterization of the free boundary.

Cite this article

L. Caffarelli, D. De Silva, O. Savin, The two membranes problem for different operators. Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 4, pp. 899–932

DOI 10.1016/J.ANIHPC.2016.05.006