Spectral Analysis of the Laplacian Operator in Two-Dimensional Strips

• Grant number: 24/01946-0
• Support Opportunities: Regular Research Grants
• Start date: July 01, 2024
• End date: June 30, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Agreement: CNPq
• Principal Investigator: Marcone Corrêa Pereira
• Grantee: Marcone Corrêa Pereira
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 20/14075-6 - Dynamical systems and their attractors under perturbations, AP.TEM
• Associated scholarship(s): 24/14305-2 - Perturbation of boundary value problems, BP.JD

Abstract

This project aims to investigate the spectrum of the Laplacian operator restricted to a two-dimensional strip embedded in a space of dimension greater than 2. Recently, this issue has been addressed under two different boundary conditions: in one situation, the Dirichlet condition was considered, while in another, a combination of Dirichlet and Neumann conditions was applied on opposite sides of the strip. Our main goal is to explore different boundary conditions to obtain new results regarding the spectrum of this class of operators, as the existence of the discrete spectrum depends on the geometry of the strip and the boundary conditions. Moreover, these conditions model a variety of distinct physical situations. Thus, this study can clarify how the spectrum of the Laplacian operator changes under those situations. For example, a condition that may lead to interesting results is the Robin boundary condition. Another situation to consider includes deformations in the boundary or the case where the strip is a periodic region. In the latter case, the spectrum of the Laplacian operator exhibits a band structure, and our objective is to study the existence and location of gaps between these bands. This is an open problem, even when considering the Dirichlet boundary condition. (AU)