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The Dirichlet problem for the convex envelope. (English) Zbl 1244.35056

Summary: The convex envelope of a given function was recently characterized as the solution of a fully nonlinear PDE [A. M. Oberman, Proc. Am. Math. Soc. 135, No. 6, 1689–1694 (2007; Zbl 1190.35107)]. 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.

MSC:

35J60 Nonlinear elliptic equations
35J70 Degenerate elliptic equations
26B25 Convexity of real functions of several variables, generalizations
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
35D40 Viscosity solutions to PDEs

Citations:

Zbl 1190.35107
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References:

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