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The Dirichlet problem for the convex envelope

Authors: Adam M. Oberman and Luis Silvestre
Journal: Trans. Amer. Math. Soc. 363 (2011), 5871-5886
MSC (2010): Primary 35J60, 35J70, 26B25, 49L25
Published electronically: June 20, 2011
MathSciNet review: 2817413
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Abstract: The convex envelope of a given function was recently characterized as the solution of a fully nonlinear partial differential equation (PDE). In this article we study a modified problem: the Dirichlet problem for the underlying PDE. The main result is an optimal regularity result. Differentiability ( $ C^{1,\alpha}$ regularity) of the boundary data implies the corresponding result for the solution in the interior, despite the fact that the solution need not be continuous up to the boundary. Secondary results are the characterization of the convex envelope as: (i) the value function of a stochastic control problem, and (ii) the optimal underestimator for a class of nonlinear elliptic PDEs.

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Additional Information

Adam M. Oberman
Affiliation: Department of Mathematics, Simon Fraser University, British Columbia, Canada

Luis Silvestre
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637

Keywords: Partial differential equations, convex envelope, viscosity solutions, stochastic control
Received by editor(s): September 27, 2008
Received by editor(s) in revised form: October 29, 2009
Published electronically: June 20, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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