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$H$-compactness for nonlocal linear operators in fractional divergence form

Maicol Caponi, Alessandro Carbotti, Alberto Maione

TL;DR

This work develops an $H$-convergence theory for nonlocal linear operators in fractional divergence form with exterior data fixed. By leveraging the Riesz potential to localize the fractional gradient, it proves $H$-compactness for the class $\mathcal{M}(\lambda,\Lambda,\Omega,A_0)$ and, in the symmetric case, $\Gamma$-compactness of the associated energies, with the $H$-limit agreeing with the local limit on $\Omega$. It then establishes an equivalence between nonlocal $H$-convergence and $\Gamma$-convergence of the energies, via a relation to local problems and a convergence of momenta. The results advance homogenization theory for fractional operators and point toward extensions to monotone operators, parabolic problems, and sub-Riemannian/nonlocal geometries.

Abstract

We study the $H$-convergence of nonlocal linear operators in fractional divergence form, where the oscillations of the matrices are prescribed outside the reference domain. Our compactness argument bypasses the failure of the classical localisation techniques that mismatch with the nonlocal nature of the operators involved. If symmetry is also assumed, we extend the equivalence between the $H$-convergence of the operators and the $Γ$-convergence of the associated energies.

$H$-compactness for nonlocal linear operators in fractional divergence form

TL;DR

This work develops an -convergence theory for nonlocal linear operators in fractional divergence form with exterior data fixed. By leveraging the Riesz potential to localize the fractional gradient, it proves -compactness for the class and, in the symmetric case, -compactness of the associated energies, with the -limit agreeing with the local limit on . It then establishes an equivalence between nonlocal -convergence and -convergence of the energies, via a relation to local problems and a convergence of momenta. The results advance homogenization theory for fractional operators and point toward extensions to monotone operators, parabolic problems, and sub-Riemannian/nonlocal geometries.

Abstract

We study the -convergence of nonlocal linear operators in fractional divergence form, where the oscillations of the matrices are prescribed outside the reference domain. Our compactness argument bypasses the failure of the classical localisation techniques that mismatch with the nonlocal nature of the operators involved. If symmetry is also assumed, we extend the equivalence between the -convergence of the operators and the -convergence of the associated energies.
Paper Structure (11 sections, 21 theorems, 255 equations, 1 figure)

This paper contains 11 sections, 21 theorems, 255 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. The functional setting
  5. The H-convergence problem
  6. The Gamma-convergence problem
  7. The equivalence problem
  8. H-compactness of nonlocal operators
  9. Gamma-compactness of nonlocal energies
  10. Equivalence between nonlocal H-convergence and Gamma-convergence
  11. Conclusions and open problems

Key Result

Proposition 2.1

Let $\alpha\in(0,n)$ and $p\in \left(1,\frac{n}{\alpha}\right)$. For every $f\in L^p(\mathbb{R}^n)$, the $\alpha$-Riesz potential $I_\alpha f$ is well-defined and there exists a positive constant $C$, depending only on $\alpha$, $n$, and $p$, such that

Figures (1)

  • Figure 1: The relations between $H$- and $\Gamma$-convergence in the case of symmetric matrices in the local and nonlocal scenarios.

Theorems & Definitions (46)

  • Proposition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.7: Leibniz rule
  • Proposition 2.8: Poincaré inequality
  • Proposition 2.9: Rellich Theorem
  • ...and 36 more