Asymptotic issue for fractional laplacian on long cylinders

TL;DR

The article analyzes how weak solutions to elliptic and parabolic problems driven by the fractional $p$-Laplacian behave as cylindrical domains expand indefinitely along one axis. By embedding the problems in a nonlocal energy framework and employing fractional Sobolev spaces, it proves convergence to reduced cross-section problems on $\\omega$ with explicit rates in terms of the expansion parameter $\\ell$. The elliptic result (Theorem THE) provides a sharp rate in $L^{p}$ on fixed subdomains under $f$ independent of the axial coordinate, while the parabolic result (Theorem THP) gives time-dependent convergence rates in $L^{\\infty}(0,T;L^{2})$ and $L^{p}(0,T;L^{p})$. These findings extend known local-domain asymptotics to the nonlocal fractional $p$-Laplacian setting and have practical implications for numerical approximations and well-posedness in unbounded domains with non-decaying data.

Abstract

In this paper, we are concerned with the asymptotic behavior of weak solutions to certain elliptic and parabolic problems involving the fractional $p$-Laplacian in cylindrical domains that become unbounded in one direction. The nonlocal nature of the operator describing the equations creates several technical difficulties in treating problems of this type. The main results, obtained within a nonlocal abstract framework, extend and complement related properties established in the local setting.

Theorems & Definitions (28)

• Theorem
• Theorem
• Definition 1.1
• Theorem 1.1
• Remark 1
• Definition 1.2
• Definition 1.3
• Theorem 1.2
• Proposition 2.1
• Lemma 2.2: Poincaré Inequality