Lower bounds for the Steklov eigenvalue problem

Authors: Davoudi, Salman
Advisor: Girouard, AlexandreMashreghi, Javad
Abstract: The Steklov problem is a spectral problem whose origin lies in the mechanics of fluids. It is an eigenvalue problem with spectral parameters in the boundary conditions, which has various applications. Its spectrum coincides with that of the Dirichlet-to-Neumann operator. The spectrum of the Steklov’s problem is discrete when the trace operator is compact. In this master’s thesis, we prove the collapse of the Steklov spectrum for a dumbbell domain in two manners. We will focus on non-Lipschitz domains. We give two examples to show that the trace operator is not compact for non-Lipschitz domains. Furthermore, we present a lower bound to the first non-zero eigenvalue σ₁ of the Steklov problem for domains having two axes of symmetry. Finally, we present lower bounds for the Steklov eigenvalue problem for starshaped domains. These results were due to J. R. Kuttler and V. G. Sigillito restrict domains to domains with two axes of symmetry or star-shaped domains [7, 8].
Document Type: Mémoire de maîtrise
Issue Date: 2019
Open Access Date: 17 April 2019
Permalink: http://hdl.handle.net/20.500.11794/34478
Grantor: Université Laval
Collection:Thèses et mémoires

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