Brezis--Kato Type Regularity Results for Higher Order Elliptic Operators

TL;DR

This paper extends Brezis--Kato regularity results to solutions of higher order nonlinear elliptic equations with variable coefficients, providing new insights into the regularity of solutions in higher dimensions.

Contribution

It establishes Brezis--Kato type regularity results for higher order elliptic operators with variable coefficients, a novel extension in elliptic PDE theory.

Findings

Proves regularity results for solutions of higher order elliptic equations.

Handles variable coefficient elliptic operators of order 2m.

Applicable to equations with nonlinearities g(x,u).

Abstract

We prove a Brezis--Kato regularity type results for solutions of the higher order nonlinear elliptic equation \[ L u = g(x,u)\qquad\text{in }\Omega \] with an elliptic operator $L$ of $2m$ order with variable coefficients and a Carath\'eodory function $g:\Omega\times \mathbb{C}\to\mathbb{C}$, where $\Omega\subset\mathbb{R}^N$ is an open set with $N > 2m$.