Topics in nonlocal and nonlinear equations
Time: Tue 2025-12-09 14.00
Location: F3, Lindstedvägen 26
Language: English
Doctoral student: Alireza Tavakoli , Matematik (Inst.), Analysis, Dynamics, Number Theory and PDE
Opponent: Professor Verena Bögelein, Paris Lodron University, Salzburg
Supervisor: Associate professor Erik Lindgren, Analys, dynamik, geometri, PDE och talteori
QC-2025-11-20
Abstract
This thesis is concerned with some qualitative properties of solutions to nonlocal equations. Nonlocal equations, as opposed to local equations such as the Laplace equation, also take into account long-range interac- tions. We are in particular interested in regularity properties, symmetry properties, and boundary behavior of solutions.The thesis includes an introduction, a summary of the results and four papers. All of the papers treat some nonlocal equation. In paper A, we study a parabolic equation involving the fractional p-Laplace op- erator with p ≥ 2. We obtain a scaling critical modulus of continuity for the equations with some right-hand side as well as a local bounded- ness result. In paper B, we study the problem of isolation of the first eigenvalue for an eigenvalue problem involving the fractional Laplace operator, which is related to a fractional Poincaré-type inequality. As a by-product, we also obtain a boundary Harnack principle. Paper C concerns a Morrey type inequality for the fractional Sobolev spaces and the associated extremal functions. We establish existence, some sym- metry properties and we show that the extremal functions have a limit at infinity. Paper D is also a study of a parabolic equation involving the fractional p-Laplace operator. In contrast to the first paper, we deal with the case p < 2. We obtain a modulus of continuity for equations with a bounded right-hand side.