{"id":{"repo_id":"bielefeld","oai_identifier":"oai:pub.uni-bielefeld.de:2962113"},"canonical_url":"https://search.dev.ndltd.org/etd/bielefeld/oai:pub.uni-bielefeld.de:2962113","repository":{"repo_id":"bielefeld","name":"Universität Bielefeld","base_url":"https://pub.uni-bielefeld.de/oai"},"display":{"title":"Regularity theory for nonlocal equations","abstract":"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.","abstract_html":"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.&lt;br /&gt;&lt;br /&gt;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.&lt;br /&gt;&lt;br /&gt;More precisely, we show that under assumptions on the right-hand side that allow for an arbitrarily small gain of integrability, weak solutions <span class=\"etd-inline-math\">u \\in W<sup>s,2</sup></span> in fact belong to <span class=\"etd-inline-math\">W<sup>t,p</sup><sub>loc</sub></span> for any $s \\leq t &lt; \\min\\{2s,1\\}$, where $p&gt;2$ reflects the amount of integrability gained. By embedding, we also obtain optimal higher Hölder regularity for such nonlocal equations.&lt;br /&gt;&lt;br /&gt;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.","abstract_has_math":true,"creators":["Nowak, Simon Noah"],"institution":"Universität Bielefeld","degree_name":null,"degree_level":"thesis.doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-03-28","date_published":"2022-03-28","updated_at":"2026-07-27T18:50:04Z","subjects":[],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://pub.uni-bielefeld.de/record/2962113","outbound_label":"Repository record","outbound_source":"source_url"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Nowak, Simon Noah"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:publisher","label":"Institution","values":["Universitätsbibliothek Bielefeld"]},{"key":"dc:type","label":"Dc Type","values":["doctoralThesis"]},{"key":"thesis:degree_level","label":"Degree Level","values":["thesis.doctoral"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Universität Bielefeld"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Regularity theory for nonlocal equations"]}]}],"canonical_facts":{"dc:creator":["Nowak, Simon Noah"],"dc:description.abstract":["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."],"dc:format.medium":["application/pdf"],"dc:publisher":["Universitätsbibliothek Bielefeld"],"dc:title":["Regularity theory for nonlocal equations"],"dc:type":["doctoralThesis"],"thesis:degree_level":["thesis.doctoral"],"thesis:institution_name":["Universität Bielefeld"]},"updated_at":"2026-07-27T18:50:04Z"}