Curved 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 centrally symmetric 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.