# A simple characterization of $H$-convergence for a class of nonlocal problems

## TL;DR

This paper provides an explicit characterization of $H$-convergence for nonlocal fractional $p$-Laplace operators, showing that weak-* convergence of coefficients is equivalent to $H$-convergence, highlighting differences from local cases.

## Contribution

It establishes that weak-* convergence of coefficients characterizes $H$-convergence for nonlocal fractional $p$-Laplace operators, extending previous work.

## Key findings

- Weak-* convergence of coefficients is equivalent to $H$-convergence.
- The result leverages nonlocality, contrasting with local $p$-Laplacian cases.
- Provides an explicit characterization of $H$-convergence for nonlocal operators.

## Abstract

This is a follow-up of a paper by Fern\'andez-Bonder-Ritorto-Salort [8], where the classical concept of $H$-convergence was extended to fractional \(p\)-Laplace type operators. In this short paper we provide an explicit characterization of this notion by demonstrating that the weak-* convergence of the coefficients is an equivalent condition for $H$-convergence of the sequence of nonlocal operators. This result takes advantage of nonlocality and is in stark contrast to the local p-Laplacian case