# American Institute of Mathematical Sciences

December  2018, 38(12): 6015-6027. doi: 10.3934/dcds.2018152

## The two membranes problem for fully nonlinear operators

 1 Mathematics Department, The University of Texas at Austin, 2515 Speedway Stop C1200, Austin, Texas 78712, USA 2 Departamento de Matemática, Universidad Nacional de Mar del Plata/Conicet, Deán Funes 3600, Mar del Plata, Buenos Aires 7600, Argentina

* Corresponding author

Received  August 2017 Revised  February 2018 Published  April 2018

We study the two membranes problem for two different fully nonlinear operators. We give a viscosity formulation for the problem and prove existence of solutions. Then we prove a general regularity result and the optimal $C^{1, 1}$ regularity when the operators are the Pucci extremal operators. We also give an example that shows that no regularity for the free boundary is to be expected to hold in general.

Citation: Luis Caffarelli, Luis Duque, Hernán Vivas. The two membranes problem for fully nonlinear operators. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6015-6027. doi: 10.3934/dcds.2018152
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