PUB - Publikationen an der Universität Bielefeld

This thesis is primarily concerned with proving Sobolev regularity results of Calderón-Zygmund-type for nonlinear nonlocal equations with possibly very irregular coefficients of VMO-type or even coefficients that are merely small in BMO. In particular, such coefficients might be discontinuous.

While for corresponding local elliptic equations with VMO coefficients it is only possible to obtain higher integrability, in our nonlocal setting we are able to also prove a substantial amount of higher differentiability. Therefore, our results are in some sense of purely nonlocal type, following a recent trend of such results in the literature.

More precisely, we show that under assumptions on the right-hand side that allow for an arbitrarily small gain of integrability, weak solutions $u \in W^{s,2}$ in fact belong to $W^{t,p}_{loc}$ for any $s \leq t < \min\{2s,1\}$, where $p>2$ reflects the amount of integrability gained. By embedding, we also obtain optimal higher Hölder regularity for such nonlocal equations.

In particular, our main results extend various previous results concerning Sobolev and Hölder regularity to the setting of nonlinear nonlocal equations with possibly discontinuous coefficients.