On a new class of fractional partial differential equations II

Abstract

In this paper we continue to advance the theory regarding the Riesz fractional gradient in the calculus of variations and fractional partial differential equations begun in an earlier work of the same name. In particular we here establish an $L^1$ Hardy inequality, obtain further regularity results for solutions of certain fractional PDE, demonstrate the existence of minimizers for integral functionals of the fractional gradient with non-linear dependence in the field, and also establish the existence of solutions to corresponding Euler-Lagrange equations obtained as conditions of minimality. In addition we pose a number of open problems, the answers to which would fill in some gaps in the theory as well as to establish connections with more classical areas of study, including interpolation and the theory of Dirichlet forms.