Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On a family of fully nonlinear integro-differential operators: From fractional Laplacian to nonlocal Monge-Ampère

Luis A. Caffarelli, María Soria-Carro

Abstract

We introduce a new family of intermediate operators between the fractional Laplacian and the Caffarelli-Silvestre nonlocal Monge-Ampère 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.

On a family of fully nonlinear integro-differential operators: From fractional Laplacian to nonlocal Monge-Ampère

Abstract

We introduce a new family of intermediate operators between the fractional Laplacian and the Caffarelli-Silvestre nonlocal Monge-Ampère 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 -regularity of solutions in the full space.
Paper Structure (10 sections, 30 theorems, 268 equations, 2 figures)

This paper contains 10 sections, 30 theorems, 268 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Construction of kernels
  3. Rearrangements and measure preserving transformations
  4. Analysis of $\mathcal{F}_k^s$
  5. Analysis of the infimum
  6. Limit as $s\to 1$
  7. Connection to optimal transport
  8. Regularity of $\mathcal{F}_k^s u$
  9. A Global Poisson Problem
  10. Future directions

Key Result

Proposition 2.2

Let $1\leq k<n$. Then $\mathcal{K}_{k}^s \subset \mathcal{K}_{k+1}^s \subseteq \mathcal{K}_{n}^s$.

Figures (2)

  • Figure 1: Area preserving deformation in $\mathbb{R}^3$.
  • Figure 2: Geometry involved in the proof of Proposition \ref{['prop:subsolution']}.

Theorems & Definitions (67)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Corollary 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • proof
  • ...and 57 more