Communications in Analysis and Geometry

Volume 30 (2022)

Number 6

Effective operators for changing sign Robin Laplacian in thin two- and three-dimensional curved waveguides

Pages: 1227 – 1268



César R. de Oliveira (Departamento de Matemática, Universidade Federal de São Carlos, SP, Brazil)

Alex F. Rossini (Instituto de Matemática, Federal University of Mato Grosso do Sul, Campo Grande, MS, Brazil)


We study the Laplacian in some thin curved domains, in the plane and space, with particular types of Robin boundary conditions and cross-sections. We derive, when the diameters of the cross sections tend to zero, nontrivial effective Schrödinger operators on the reference curve by means of norm resolvent convergences. Besides the changing sign in the Robin parameter, for which no renormalization is necessary, another novelty is that the torsion (in the spatial case) plays no role to effective operators.

Received 18 July 2016

Accepted 21 November 2019

Published 26 April 2023