Stability results for nonlocal Serrin-type problems, antisymmetric Harnack inequalities, and geometric estimates

TL;DR

This thesis investigates symmetry, stability, and geometric properties of nonlocal PDEs, focusing on overdetermined problems, antisymmetric Harnack inequalities, and fractional mean curvature.

Contribution

It provides new stability results, geometric identities, and inequalities for nonlocal PDEs, expanding understanding of their symmetry and geometric behavior.

Findings

Stability results for nonlocal Serrin-type problems

Antisymmetric Harnack inequalities for nonlocal PDE solutions

Geometric identities involving fractional mean curvature

Abstract

In this thesis, we explore several related topics broadly regarding the symmetry and geometric properties of nonlocal partial differential equations (PDE). This thesis is split into three parts. In the first part, we study two overdetermined problems, namely Serrin's problem and the parallel surface problem, driven by the fractional Laplacian. In the second part, we study the Harnack inequality for solutions to nonlocal PDE which are antisymmetric, that is, they have an odd symmetry with respect to reflections across some hyperplane. This topic has a strong motivation coming from proving quantitative stability estimates for nonlocal overdetermined problems. In the third part, we prove several geometric identities and inequalities involving the fractional mean curvature.