On Axially Symmetric Solutions of Fully Nonlinear Elliptic Equations

TL;DR

This paper proves that in three dimensions, axially symmetric viscosity solutions to fully nonlinear elliptic Hessian equations are actually classical solutions, confirming their smoothness under symmetry assumptions.

Contribution

It establishes the regularity of axially symmetric viscosity solutions for fully nonlinear elliptic Hessian equations in three dimensions.

Findings

Viscosity solutions are classical in the symmetric setting.

Regularity results hold specifically in three dimensions.

Symmetry plays a key role in the smoothness of solutions.

Abstract

We show that in dimension 3 axial-symmetric viscosity solutions of uniformly elliptic Hessian equations are in fact the classical ones.