Fractional differentiability for solutions of nonlinear elliptic equations

TL;DR

This paper investigates the fractional differentiability of solutions to nonlinear elliptic equations with divergence form, establishing local well-posedness in Besov spaces under certain smoothness and growth conditions.

Contribution

It introduces new regularity results for solutions of nonlinear elliptic equations with fractional differentiability, extending previous understanding to equations with variable coefficients and growth conditions.

Findings

Solutions exhibit fractional differentiability in Besov spaces under specified conditions.

Local well-posedness is established for equations with coefficients in Besov and VMO spaces.

Results apply to equations with linear and certain nonlinear growth in the gradient.

Abstract

We study nonlinear elliptic equations in divergence form $${\operatorname{div}}{\mathcal A}(x,Du)={\operatorname{div}}G.$$ When ${\mathcal A}$ has linear growth in $Du$, and assuming that $x\mapsto{\mathcal A}(x,\xi)$ enjoys $B^\alpha_{\frac{n}\alpha, q}$ smoothness, local well-posedness is found in $B^\alpha_{p,q}$ for certain values of $p\in[2,\frac{n}{\alpha})$ and $q\in[1,\infty]$. In the particular case ${\mathcal A}(x,\xi)=A(x)\xi$, $G=0$ and $A\in B^\alpha_{\frac{n}\alpha,q}$, $1\leq q\leq\infty$, we obtain $Du\in B^\alpha_{p,q}$ for each $p<\frac{n}\alpha$. Our main tool in the proof is a more general result, that holds also if ${\mathcal A}$ has growth $s-1$ in $Du$, $2\leq s\leq n$, and asserts local well-posedness in $L^q$ for each $q>s$, provided that $x\mapsto{\mathcal A}(x,\xi)$ satisfies a locally uniform $VMO$ condition.