Some nonlocal operators in porous medium equations: the extension problem and regularity theory

Abstract

The value of integro-differentlal (non!oca!) operators to nonlinear differential equations is we!l-known. The combination of integro-differential operators and porous medium nonlinearities gives rise to interesting mathematical models that have been studied in the last decade both because of their mathematical properties and a number of scientific applications in different fields, such as engineering, physics, medicine and biology. ln this PhD Thesis, we are concerned with a number of nonlocal variants and extensions of nonlinear differential equations of porous medium-types, with suitable functional inequalities associated with the underlying functional spaces. We address several fundamenta! issues both linear and non!inear such as existence, uniqueness, boundary and interior regularity for nonnegative solutions. The De Giorgi-Nash-Moser HOider regularity techniques are used to prove continuity of solutions.