A Dirichlet Problem in a Domain with a Small Hole

Matteo Dalla Riva, Massimo Lanza de Cristoforis and Paolo Musolino
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Matteo Dalla Riva: The University of Tulsa, College of Engineering and Natural Science
Massimo Lanza de Cristoforis: Università degli Studi di Padova, Dipartimento di Matematica
Paolo Musolino: Università Ca’ Foscari Venezia, Dipartimento di Scienze Molecolari e Nanosistemi

Chapter Chapter 8 in Singularly Perturbed Boundary Value Problems, 2021, pp 261-335 from Springer

Abstract: Abstract In this chapter we introduce the Functional Analytic Approach for the analysis of elliptic boundary value problems in singularly perturbed domains. To illustrate how the method works, we closely study a boundary value problem for the Laplace equation in a bounded domain with a small hole that shrinks to an interior point. We impose Dirichlet conditions both on the outer boundary of the domain and on the boundary of the hole. One peculiarity of this specific problem is that, in the case of dimension two, the solution displays a logarithmic behavior. We shall see in Chapter 9 that such behavior might not appear with other boundary conditions.

Date: 2021
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DOI: 10.1007/978-3-030-76259-9_8

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