Abstract:
Integro-differential operators arise naturally in biological modeling and mathematical finance. We aim to conduct an in-depth study of integro-differential operators and their regularity properties in this thesis. We start by considering linear integro-differential operators of L\'evy type and by studying existence-uniqueness results for the associated boundary-value problems, maximum principles, and generalized eigenvalue problems. As an application of these results, we discuss Faber-Krahn inequality and a one-dimensional symmetry result related to the Gibbons' conjecture.
Next we bring our attention to the boundary regularity of the solutions of linear integro-differential operators over bounded domains and we prove that these solutions are globally $C^{1, \alpha}$ regular. This is also used to study an overdetermined problem. To extend the linear case, we consider fully nonlinear, non-translation invariant integro-differential operators and discuss boundary regularity of solutions which requires a careful construction of a sub and supersolutions and appropriate Harnack type inequality.
At last, we consider fully nonlinear nonlocal operators. We establish H\"{o}lder regularity, Harnack inequality and boundary Harnack estimates. As an application of maximum principles, regularity theory and generalized eigenvalue problems, we then discuss one of the most celebrated reaction-diffusion model, known in literature as Fisher-KPP model, in the nonlocal setting. We further establish the existence, uniqueness and multiplicity results of the solutions to the steady state Fisher-KPP equation and long time asymptotic of the solutions of the parabolic counterpart.
