Event Details
Speaker: M. Kassmann (Bielefeld)
Abstract: We discuss (a) well-posedness and (b) regularity for variational solutions to nonlocal equations in bounded domains. We focus on nonlocal effects due to (a) the integration across the boundary and (b) the presence of long-range tail terms.
(a) We study function spaces and provide robust nonlocal trace and extension theorems. The trace and extension results for Sobolev-type function spaces are well suited for nonlocal Dirichlet and Neumann problems including those for the fractional p-Laplacian. Here, the focus is on the choice of exterior data. We prove results that are robust with respect to the order of differentiability. In this sense they are in align with the classical trace and extension theorems. Then we discuss the nonlocal-to-local transition. We show convergence of solutions of parametrized nonlocal problems to limiting classical local problems. The first three lectures are based on the following works:
- https://arxiv.org/pdf/2204.06793 (joint with Guy Foghem)
- https://arxiv.org/pdf/2305.05735 (joint with Florian Grube)
- https://arxiv.org/pdf/2209.04397 (Florian Grube/Thorben Hensiek)
A proof of the classical divergence theorem based on the notion of nonlocal divergence and some approximation can be found here:
- https://arxiv.org/pdf/2302.01945 (joint with Solveig Hepp)
The last lecture addresses Hölder regularity and Harnack's inequality for solutions to elliptic and parabolic nonlocal problems. We focus on challenges created by tail terms. The lecture reviews results of the last 20 years and explains the contributions of the recent work:
https://arxiv.org/pdf/2303.05975.pdf (joint with Marvin Weidner)
