Existence of minimizers of functionals involving the fractional gradient in the abscence of compactness, symmetry and monotonicity

TL;DR

This paper proves the existence of minimizers for certain fractional gradient functionals under broad conditions, even without compactness, symmetry, or monotonicity assumptions.

Contribution

It introduces general conditions ensuring minimizers for fractional gradient functionals in constrained variational problems without relying on traditional compactness or symmetry.

Findings

Minimizers exist under broad assumptions.

Applicable to problems lacking symmetry or monotonicity.

Extends variational methods to fractional gradient functionals.

Abstract

We establish general assumptions under which a constrained vari- ational problem involving the fractional gradient and a local nonlin- earity admits minimizers.