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A Dirichlet Problem in a Domain with Two Small Holes

In: Singularly Perturbed Boundary Value Problems

Author

Listed:
  • 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)

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.

Suggested Citation

  • Matteo Dalla Riva & Massimo Lanza de Cristoforis & Paolo Musolino, 2021. "A Dirichlet Problem in a Domain with Two Small Holes," Springer Books, in: Singularly Perturbed Boundary Value Problems, chapter 0, pages 373-431, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-76259-9_10
    DOI: 10.1007/978-3-030-76259-9_10
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