The Dirichlet problem for the convex envelope

By Luis Silvestre, Adam M. Oberman. 2010

74 citations recorded by the source corpus.

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 (C1,α 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/e3018d7b57296f28128ce9ec10fe34722316f8bf

Citations in the corpus

• Games for eigenvalues of the Hessian and concave/convex envelopes (2018, unknown): “See [12].”
• A family of degenerate elliptic operators: Maximum principle and its consequences (2016, unknown): “Still concerning the regularity, let us recall that in the context of convex analysis, Oberman and Silvestre in [25] prove the C1,α regularity of solutions of P 1 (D (2)u) = 0 in Ω, u(x) = g(x) on ∂Ω, under some regularity condition on g.”