Singular Solutions of Hessian Fully Nonlinear Elliptic Equations

TL;DR

This paper investigates the regularity and singularity of solutions to Hessian fully nonlinear elliptic equations, demonstrating that solutions can exhibit interior blow-up and establishing the optimal regularity bounds in high dimensions.

Contribution

It proves the existence of solutions with interior singularities in dimensions 11 and higher, and establishes the optimal interior regularity bounds for such equations.

Findings

Solutions can blow up inside the domain in 12 or more dimensions.

Optimal interior regularity is at most C^{1+ε}.

Existence of non-smooth solutions in 11 dimensions.

Abstract

We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C^{1+\epsilon}, showing the optimality of the known interior regularity result. The same is proven for Isaacs equations. We prove the existence of non-smooth solutions to fully nonlinear Hessian uniformly elliptic equations in 11 dimensions. We study also the possible singularity of solutions of Hessian equations defined in a neighborhood of a point and prove that a homogeneous order 0<\alpha<1 solution of a Hessian uniformly elliptic equation in a punctured ball should be radial.