On a new class of fractional partial differential equations II

TL;DR

This paper advances the mathematical theory of fractional PDEs by establishing inequalities, regularity results, and existence theorems for solutions involving the Riesz fractional gradient, connecting fractional calculus with classical analysis.

Contribution

It introduces new regularity results, existence proofs for minimizers and solutions, and poses open problems linking fractional PDEs to classical analysis areas.

Findings

Established an $L^1$ Hardy inequality for fractional gradients

Proved existence of minimizers for nonlinear fractional variational problems

Demonstrated solutions to fractional Euler-Lagrange equations

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.