A Dirichlet Problem in a Domain with Two Small Holes

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 10 in Singularly Perturbed Boundary Value Problems, 2021, pp 373-431 from Springer

Abstract: Abstract In this chapter we see an application of the Functional Analytic Approach to a domain perturbation of a somewhat different nature: instead of a domain with a single shrinking hole we consider a domain that contains two holes that collide into one another while shrinking in size. We confine ourselves to a Dirichlet problem for the Laplace equation. Different boundary value problems can also be analyzed and some examples can be found in Dalla Riva and Musolino (Comm Partial Differ Equ 41(5):812–837, 2016; Math Methods Appl Sci 41(3):986–993, 2018). Yet as occurs for the single hole problem of Chapter 8 , the Dirichlet problem presents certain interesting features related to the appearance of a logarithmic behavior in the two-dimensional case. For this reason we prefer to focus on this problem and investigate it thoroughly.

Date: 2021
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DOI: 10.1007/978-3-030-76259-9_10

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