$BMO$ and gradient estimates for solutions of critical elliptic equations

TL;DR

This paper investigates the regularity of solutions to critical elliptic equations using spaces of bounded mean oscillation, extending classical results to more general function spaces and establishing new gradient estimates.

Contribution

It introduces applications of $eta$-dimensional BMO spaces to elliptic regularity, improving classical results and providing new gradient bounds for solutions.

Findings

Functions with gradient in weak-$L^n$ are in $BMO^eta$ for all $eta o n$

Solutions to Poisson and $n$-Laplace equations are in $BMO^eta$ under weaker conditions

New gradient estimate for $n$-Laplace equations with divergence-free data

Abstract

In this paper we explore several applications of the recently introduced spaces of functions of bounded $\beta$-dimensional mean oscillation for $\beta \in (0,n]$ to regularity theory of critical exponent elliptic equations. We first show that functions with gradient in weak-$L^n$ are in $BMO^\beta$ for any $\beta \in (0,n]$, improving the classical result $\nabla u\in L^n$ implies $u\in BMO$. We apply this result to the Poisson equation $-\Delta u = \operatorname*{div} F$ with zero boundary conditions in a bounded $C^1$ domain to show that $u\in BMO^{\beta}$ when $F$ is in weak-$L^n$. Next, we consider the $n$-Laplace equation \begin{align*} -\operatorname*{div}( |\nabla U|^{n-2} \nabla U) &= F \text{ in } \Omega, \newline U &=0 \text{ on }\partial \Omega. \end{align*} with $F\in L^1(\Omega)$ and show that the classical result $u\in BMO$ can be improved to $u\in BMO^\beta$. Finally, we consider the $n$-Laplace equation in the case when $F \in L^1$, $\operatorname*{div} F=0$ and prove that for smooth domains $\Omega$ we have the estimate \begin{align*} \|\nabla U \|_{L^n} \mathbb \leq C \, \|F\|^{1/(n-1)}_{L^1}, \end{align*} where the constant $C$ is independent of $F$.