Fractional differentiability for solutions of nonlinear elliptic equations

Abstract

We study nonlinear elliptic equations in divergence form $$\mathrm{div}\mathcal{A}(x,Du)=\mathrm{div}G.$$ When $\mathcal{A}$ has linear growth in $Du$, and assuming that $x\mapsto \mathcal{A}(x,\xi )$ enjoys $B_{\frac{n}{\alpha },q}^{\alpha }$ smoothness, local well-posedness is found in $B_{p,q}^{\alpha }$ 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 $A\in B_{\frac{n}{\alpha },q}^{\alpha }$, $1\le q\le \infty$, we obtain $Du\in B_{p,q}^{\alpha }$ 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 $Du$, $2\le s\le n$, 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.