Asymptotic analysis of the Dirichlet fractional Laplacian in domains becoming unbounded

TL;DR

This paper investigates how the Dirichlet fractional Laplacian behaves asymptotically on domains that expand infinitely in certain directions, revealing a dimension reduction phenomenon through Gamma-convergence analysis.

Contribution

It introduces a novel analysis of the fractional Laplacian on unbounded domains, demonstrating a dimension reduction via Gamma-convergence.

Findings

Dimension reduction phenomenon observed

Gamma-convergence describes asymptotic behavior

Provides insights into fractional Laplacian on unbounded domains

Abstract

In this paper we analyze the asymptotic behavior of the Dirichlet fractional Laplacian $(-\Delta_{\mathbb R^{n+k}})^{s}$, with $s\in (0, 1)$, on bounded domains in $\mathbb R^{n+k}$ that become unbounded in the last $k$-directions. A dimension reduction phenomenon is observed and described via $\Gamma$-convergence.