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On a family of fully nonlinear integrodifferential operators: from fractional Laplacian to nonlocal Monge–Ampère

Luis A. Caffarelli and María Soria-Carro

Vol. 17 (2024), No. 1, 243–279
Abstract

We introduce a new family of intermediate operators between the fractional Laplacian and the nonlocal Monge–Ampère introduced by Caffarelli and Silvestre that are given by infimums of integrodifferential 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 C1,1-regularity of solutions in the full space.

Keywords
nonlinear elliptic equations, integrodifferential operators, fractional Laplacian, Monge–Ampère, rearrangements
Mathematical Subject Classification
Primary: 35J60
Secondary: 35B65, 35J96
Milestones
Received: 1 December 2021
Revised: 5 May 2022
Accepted: 16 June 2022
Published: 5 February 2024
Authors
Luis A. Caffarelli
Department of Mathematics
The University of Texas at Austin
Austin, TX
United States
María Soria-Carro
Department of Mathematics
Rutgers University
Piscataway, NJ
United States
Department of Mathematics
The University of Texas at Austin
Austin, TX
United States

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