Local regularity for concave homogeneous complex degenerate elliptic equations comparable to the Monge-Amp\`ere equation

TL;DR

This paper proves local regularity for solutions to complex degenerate elliptic equations similar to the Monge-Ampère equation, including applications to the k-Monge-Ampère equation, advancing understanding of their smoothness properties.

Contribution

It establishes a local regularity result for $W^{2,p}_{loc}$ solutions to complex degenerate elliptic equations comparable to the Monge-Ampère equation, including the $k$-Monge-Ampère case.

Findings

Proves $W^{2,p}_{loc}$ regularity for solutions to complex degenerate elliptic equations.

Applies the regularity result to the $k$-Monge-Ampère equation.

Enhances understanding of regularity in complex nonlinear elliptic equations.

Abstract

In this paper, we establish a local regularity result for $W^{2,p}_{\mathrm{loc}}$ solutions to complex degenerate nonlinear elliptic equations $F(D^2_{\mathbb{C}} u)=f$ when they are comparable to the Monge-Amp\`ere equation. Notably, we apply our result to the so-called $k$-Monge-Amp\`ere equation.