Curved nonlinear waveguides

TL;DR

This work extends the spectral-geometric analysis of quantum waveguides to the nonlinear Dirichlet p-Laplacian on tubes around unbounded curves in $\mathbb{R}^d$. By employing curvilinear coordinates and variational methods, it establishes (i) stability of the essential spectrum to the cross-section threshold for asymptotically straight tubes, (ii) a bending-induced reduction of the spectral threshold for nontrivially bent, untwisted circular cross-sections, and (iii) a robust Hardy inequality in the unbent twisted setting that holds in all dimensions $d\ge3$ and for $p>1$. The results provide a nonlinear generalization of known linear outcomes, clarifying how bending and twisting shape the spectrum of the $p$-Laplacian in curved tubular domains and offering new tools for understanding spectral stability and transport in nonlinear waveguides.

Abstract

The Dirichlet p-Laplacian in tubes of arbitrary cross-section along infinite curves in Euclidean spaces of arbitrary dimension is investigated. First, it is shown that the gap between the lowest point of the generalised spectrum and the essential spectrum is positive whenever the cross-section is circular and the tube is asymptotically straight, untwisted and non-trivially bent. Second, a Hardy-type inequality is derived for unbent and non-trivially twisted tubes.

Theorems & Definitions (25)

• Theorem 1: stability of $\lambda_\infty$
• Theorem 2: bending
• Corollary 1
• Theorem 3: twisting
• Remark 1
• Lemma 1
• proof
• Proposition 1
• proof
• Proposition 2