STABILITY RESULTS FOR NONLOCAL SERRIN-TYPE PROBLEMS, ANTISYMMETRIC HARNACK INEQUALITIES AND GEOMETRIC ESTIMATES

In this thesis, we explore several related topics broadly regarding the symmetry and geometric properties of nonlocal partial differential equations (PDE).Nonlocal PDE, also known as integro-differential equations or fractional PDE, have become increasingly popular over the past few decades due to their ability to accurately model real-world phenomena.Nonlocal PDE arise naturally in the study of stochastic processes and can be viewed as generalisations of PDE which allow the possibility of long-range interactions or forces.Some examples where nonlocal PDE arise in the applied sciences are crystal dislocation, elasticity, ecology, mathematical finance, the surface quasigeostrophic equation in fluid mechanics, image processing, the physics of plasmas and flames, magnetised plasmas and quantum mechanics.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 Chapter 2, we use a Hopf-type lemma for antisymmetric supersolutions to the Dirichlet problem for the fractional Laplacian with zeroth-order terms, in combination with the method of moving planes, to prove symmetry for the semilinear fractional parallel surface problem.That is, we prove that nonnegative solutions to semilinear Dirichlet problems for the fractional Laplacian in a bounded open set R n must be radially symmetric if one of their level surfaces is parallel to the boundary of ; in turn, must be a ball.