Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE

Robert Jensen; Andrzej Świech

doi:10.3934/cpaa.2005.4.187

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2005, Volume 4, Issue 1: 199-207. Doi: 10.3934/cpaa.2005.4.187

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Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE

Robert Jensen^1, and
Andrzej Świech^2,

1.
Department of Mathematical Sciences, Loyola University of Chicago, 6325 North Sheridan Road, Chicago, IL 60626
2.
School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332-0160

Received: February 29, 2004

Revised: September 30, 2004

Published: February 2005

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Abstract

Two results are proved in the paper. The first is a uniqueness theorem for viscosity solutions of Dirichlet boundary value problems for Bellman-Isaacs equations with just measurable lower order terms. The second is a proof that there always exist maximal and minimal viscosity solutions of Dirichlet boundary value problems for fully nonlinear, uniformly elliptic PDE that are measurable in the $x$-variable.

Keywords:

Fully nonlinear elliptic PDE,
viscosity solutions.

Mathematics Subject Classification: 35J60, 35J65, 35J25, 49L25.

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