Representation and regularity for the Dirichlet problem for real and complex degenerate Hessian equations

TL;DR

This paper proves interior regularity results for solutions to degenerate Hessian equations, extending understanding of their behavior under natural geometric conditions in both real and complex settings.

Contribution

It establishes optimal interior $C^{1,1}$ regularity for solutions to degenerate Hessian equations under geometric conditions, using a unified Bellman equation approach.

Findings

Interior $C^{1,1}$ regularity of solutions is achieved.

Results hold for both real and complex Hessian equations.

Regularity is optimal even with smooth boundary data.

Abstract

We consider the Dirichlet problem for positively homogeneous, degenerate elliptic, concave (or convex) Hessian equations. Under natural and necessary conditions on the geometry of the domain, with the $C^{1,1}$ boundary data, we establish the interior $C^{1,1}$-regularity of the unique (admissible) solution, which is optimal even if the boundary data is smooth. Both real and complex cases are studied by the unified (Bellman equation) approach.