Stochastic homogenization of fully nonlinear uniformly elliptic equations revisited

Solvability of uniformly elliptic fully nonlinear PDE

We get existence, uniqueness and non-uniqueness of viscosity solutions of uniformly elliptic fully nonlinear equations of Hamilton-Jacobi-BellmanIsaacs type, with unbounded ingredients and quadratic growth in the gradient, without hypotheses of convexity or properness. Some of our results are new even for equations in divergence form.

Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations

Perturbing Fully Nonlinear Second Order Elliptic Equations

We present two types of perturbations with reverse effects on some scalar fully nonlinear second order elliptic differential operators: on the other hand, first order perturbations which destroy the global solvability of the Dirichlet problem, in smooth bounded domains of Rn; on the other hand, an integral perturbation which restore the local solvability, on compact connected manifolds without ...

Overdetermined problems for fully nonlinear elliptic equations

We prove that the existence of a solution to a fully nonlinear elliptic equation in a bounded domain Ω with an overdetermined boundary condition prescribing both Dirichlet and Neumann constant data forces the domain Ω to be a ball. This is a generalization of Serrin’s classical result from 1971.

Degenerate Conformally Invariant Fully Nonlinear Elliptic Equations

There has been much work on conformally invariant fully nonlinear elliptic equations and applications to geometry and topology. See for instance [17], [5], [4], [10], [14], [9], and the references therein. An important issue in the study of such equations is to classify entire solutions which arise from rescaling blowing up solutions. Liouville type theorems for general conformally invariant fu...