Paolo Musolino, Martin Dutko, Gennady Mishuris
Abstract:
We consider a perforated domain \(\Omega(\epsilon)\) of \(\mathbb{R}^2\) with
a small hole of size \(\epsilon\) and we study the behavior of the solution
of a mixed Neumann-Robin problem in \(\Omega(\epsilon)\) as the size
\(\epsilon\) of the small hole tends to \(0\). In addition to the geometric degeneracy of the problem, the nonlinear \(\epsilon\)-dependent Robin
condition may degenerate into a Neumann condition for \(\epsilon=0\) and the Robin datum may diverge to infinity.
Our goal is to analyze the asymptotic behavior of the solutions to the
problem as \(\epsilon\) tends to \(0\) and to understand how the boundary condition affects the behavior of the solutions when \(\epsilon\) is close to \(0\). The present paper extends to the planar case the results of [36]
dealing with the case of dimension \(n\geq 3\).
Submitted February 15, 2023. Published September 11, 2023.
Math Subject Classifications: 35J25, 31B10, 35B25, 35C20, 47H30.
Key Words: Singularly perturbed boundary value problem; Laplace equation; nonlinear Robin condition; perforated planar domain; integral equation
DOI: 10.58997/ejde.2023.57
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Paolo Musolino Dipartimento di Scienze Molecolari e Nanosistemi Universit\`a Ca' Foscari Venezia via Torino 155, 30172 Venezia Mestre, Italy email: paolo.musolino@unive.it |
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Martin Dutko Rockfield Software Limited King's Road, Ethos Building Swansea, SA1 8PH, Wales UK email: martin.dutko@rockfieldglobal.com |
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Gennady Mishuris Department of Mathematics Aberystwyth University Ceredigion, Aberystwyth, SY23 3BZ Wales, UK email: ggm@aber.ac.uk |
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