Regularity theory and fractionally convex functions

Abstract

In this thesis, we study the regularity theory developed for uniformly elliptic operators and adapt some of the techniques to the fractional convex operator, as defined in [8],. The de generacy and nonlocal nature of the fractional convex operator posses significant challenges, which limits the extent to which the theory of the uniform elliptic regularity can be adapted to this operator. We first study the regularity theory developed for local and nonlocal opera tors, and showcase the importance of fractional convexity. Then, we describe the properties of the fractionally convex functions.