Abstract. We provide a deterministic-control-based interpretation for a broad class of fully nonlinear parabolic and elliptic PDEs with continuous Neumann boundary conditions in a smooth domain. We construct families of two-person games depending on a small parameter ε which extend those proposed by Kohn and Serfaty [21]. These new games treat a Neumann boundary condition by introducing some specific rules near the boundary. We show that the value function converges, in the viscosity sense, to the solution of the PDE as ε tends to zero. Moreover, our construction allows us to treat both the oblique and the mixed type Dirichlet-Neumann boundary conditions.
Abstract. For a parabolic equation associated to a uniformly elliptic operator, we obtain a W 3,ε estimate, which provides a lower bound on the Lebesgue measure of the set on which a viscosity solution has a quadratic expansion. The argument combines parabolic W 2,ε estimates with a comparison principle argument. As an application, we show, assuming the operator is C 1 , that a viscosity solution is C 2,α on the complement of a closed set of Hausdorff dimension ε less than that of the ambient space, where the constant ε > 0 depends only on the dimension and the ellipticity.
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