On the Dirichlet problem for the degenerate $k$-Hessian equation

TL;DR

This paper establishes optimal regularity conditions on the right-hand side function for the existence of global solutions to the degenerate $k$-Hessian Dirichlet problem, extending known results for the Monge-Ampère case.

Contribution

It identifies and proves the optimal regularity conditions on $f$ for the existence of $C^{1,1}$ solutions to the degenerate $k$-Hessian equation, generalizing previous results.

Findings

Optimal regularity conditions for $f$ are established.

Existence results are extended to the degenerate $k$-Hessian equation.

Conditions are shown to be sharp via classical counterexamples.

Abstract

This paper investigates the existence of a global $C^{1,1}$ solution to the Dirichlet problem for the $k$-Hessian equation with a nonnegative right-hand side $f$, focusing on the required conditions for $f$. The conditions $f^{1/(k-1)}\in C^{1,1}(\overline{\Omega_{0}})$ and $f^{3/(2k-2)}\in C^{2,1}(\overline{\Omega_{0}})$, together with $f\geq0$ in a domain $\Omega_{0}\Supset\Omega$, are optimal, as demonstrated by classical counterexamples. For the Monge-Amp\`ere equation ($k=n$), we establish the existence under the optimal condition $f^{3/(2n-2)}\in C^{2,1}(\overline{\Omega_{0}})$ together with $f\geq0$ in $\Omega_{0}$. For the general $k$-Hessian equation, we establish the existence under the condition $f\geq0$ in $\Omega_{0}$ together with one of the following three conditions: \begin{align*} &(1)\quad f^{1/(k-1)}\in C^{1,1}(\overline{\Omega_{0}}),\ \ \inf_{\Omega}\Delta u\geq1,\ \ 2\leq k\leq n-1;\\ &(2)\quad f^{3/(2k-2)}\in C^{2,1}(\overline{\Omega_{0}}),\ \ \inf_{\Omega}\Delta u\geq1,\ \ 5\leq k\leq n-1;\\ &(3)\quad f^{3/(2k)}\in C^{2,1}(\overline{\Omega_{0}}),\ \ 2\leq k\leq n-1. \end{align*}