Fractional Differentiability for Solutions of Nonlinear Elliptic Equations

Abstract

We study nonlinear elliptic equations in divergence form

$\text {div }{\mathcal A}(x,Du)=\text {div } G.$

When ${\mathcal A}$ has linear growth in D u, 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 D u, 2 ≤ s ≤ 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.