On a family of fully nonlinear integro-differential operators: From fractional Laplacian to nonlocal Monge-Amp\`ere

TL;DR

This paper introduces a new family of nonlocal operators bridging the fractional Laplacian and nonlocal Monge-Ampère, establishing their properties, connections to optimal transport, and analyzing solutions to related Poisson problems.

Contribution

It defines a novel class of integro-differential operators, explores their representations and links to optimal transport, and proves existence, uniqueness, and regularity of solutions for associated Poisson problems.

Findings

Established representation formulas for the new operators

Connected the operators to optimal transport theory

Proved existence, uniqueness, and regularity of solutions

Abstract

We introduce a new family of intermediate operators between the fractional Laplacian and the Caffarelli-Silvestre nonlocal Monge-Amp\`ere that are given by infimums of integro-differential operators. Using rearrangement techniques, we obtain representation formulas and give a connection to optimal transport. Finally, we consider a global Poisson problem, prescribing data at infinity, and prove existence, uniqueness, and $C^{1,1}$-regularity of solutions in the full space.