Electron. J. Differential Equations, Vol. 2023 (2023), No. 57, pp. 1-21.

Asymptotic analysis of perturbed Robin problems in a planar domain

Paolo Musolino, Martin Dutko, Gennady Mishuris

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

Show me the PDF file (372 KB), TEX file for this article.

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
Martin Dutko
Rockfield Software Limited
King's Road, Ethos Building
Swansea, SA1 8PH, Wales UK
email: martin.dutko@rockfieldglobal.com
Gennady Mishuris
Department of Mathematics
Aberystwyth University
Ceredigion, Aberystwyth, SY23 3BZ Wales, UK
email: ggm@aber.ac.uk

Return to the EJDE web page