The Dirichlet problem for the convex envelope
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- by Adam M. Oberman and Luis Silvestre PDF
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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.References
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Additional Information
- Adam M. Oberman
- Affiliation: Department of Mathematics, Simon Fraser University, British Columbia, Canada
- MR Author ID: 667376
- Email: aoberman@sfu.ca
- Luis Silvestre
- Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
- MR Author ID: 757280
- Email: luis@math.uchicago.edu
- Received by editor(s): September 27, 2008
- Received by editor(s) in revised form: October 29, 2009
- Published electronically: June 20, 2011
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 5871-5886
- MSC (2010): Primary 35J60, 35J70, 26B25, 49L25
- DOI: https://doi.org/10.1090/S0002-9947-2011-05240-2
- MathSciNet review: 2817413