Fractional differentiability for solutions of nonlinear elliptic equations

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 $x\mapsto{\mathcal A}(x,ξ)$ enjoys $B^α_{\frac{n}α, q}$ smoothness, local well-posedness is found in $B^α_{p,q}$ for certain values of $p\in[2,\frac{n}α)$ and $q\in[1,\infty]$. In the particular case ${\mathcal A}(x,ξ)=A(x)ξ$, $G=0$ and $A\in B^α_{\frac{n}α,q}$, $1\leq q\leq\infty$, we obtain $Du\in B^α_{p,q}$ for each $p<\frac{n}α$. 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 $x\mapsto{\mathcal A}(x,ξ)$ satisfies a locally uniform $VMO$ condition.

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