Regularity theory for nonlocal equations

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.<br /><br />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.<br /><br />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.<br /><br />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.

Level thesis:degree_level

thesis.doctoral

Grantor dc:publisher

Universität Bielefeld

Year

2022

Author dc:creator

• Nowak, Simon Noah

Repository record source_url

https://pub.uni-bielefeld.de/record/2962113

OAI identifier oai:identifier

oai:pub.uni-bielefeld.de:2962113