The two membranes problem for fully nonlinear operators

Luis Caffarelli; Luis Duque; Hernán Vivas

doi:10.3934/dcds.2018152

Article Contents

2018, Volume 38, Issue 12: 6015-6027. Doi: 10.3934/dcds.2018152

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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
* Corresponding author

Received: August 2017

Revised: February 2018

Published: December 2018

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Abstract

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.

Keywords:

Free boundary problems,
fully nonlinear.

Mathematics Subject Classification: 35R35, 35J60.

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References

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