On the Asymptotic Analysis of Problems Involving Fractional Laplacian in Cylindrical Domains Tending to Infinity

TL;DR

This paper investigates the asymptotic behavior of fractional Laplacian problems in unbounded cylindrical domains, extending existing results to all fractional orders between 0 and 1, and providing new weighted estimates.

Contribution

It extends the asymptotic analysis of fractional Laplacian problems to all orders between 0 and 1 and generalizes weighted estimates for solutions in unbounded domains.

Findings

Proved the non-local analogue of classical elliptic problem results.

Extended weighted estimates to fractional orders between 0 and 1.

Demonstrated asymptotic behavior in unbounded cylindrical domains.

Abstract

The article is an attempt to investigate the issues of asymptotic analysis for problems involving fractional Laplacian where the domains tend to become unbounded in one-direction. Motivated from the pioneering work on second order elliptic problems by Chipot and Rougirel, where the force functions are considered on the cross section of domains, we prove the non-local counterpart of their result. Furthermore, recently Yeressian established a weighted estimate for solutions of nonlocal Dirichlet problems which exhibit the asymptotic behavior. The case whens= 1=2 was also treated as an example to show how the weighted estimate might be used to achieve the asymptotic behavior. In this article, we extend this result to each order between 0 and 1.