We consider a Dirichlet problem in a planar domain with a hole of diameter proportional to a real parameter $epsilon$ and we denote by $u_epsilon$ the corresponding solution. The behavior of $u_epsilon$ for $epsilon$ small and positive can be described in terms of real analytic functions of two variables evaluated at $(epsilon,1/logepsilon)$. We show that under suitable assumptions on the geometry and on the boundary data one can get rid of the logarithmic behavior displayed by $u_epsilon$ for $epsilon$ small and describe $u_epsilon$ by real analytic functions of $epsilon$. Then it is natural to ask what happens when $epsilon$ is negative. The case of boundary data depending on $epsilon$ is also considered. The aim is to study real analytic families of harmonic functions which are not necessarily solutions of a particular boundary value problem.
We consider a Dirichlet problem in a planar domain with a hole of diameter proportional to a real parameter ε and we denote by uε the corresponding solution. The behavior of uε for ε small and positive can be described in terms of real analytic functions of two variables evaluated at (ε, 1/log ε). We show that under suitable assumptions on the geometry and on the boundary data one can get rid of the logarithmic behavior displayed by uε for ε small and describe uε by real analytic functions of ε. Then it is natural to ask what happens when ε is negative. The case of boundary data depending on ε is also considered. The aim is to study real analytic families of harmonic functions which are not necessarily solutions of a particular boundary value problem.
Autori: | ||
Data di pubblicazione: | 2015 | |
Titolo: | Real analytic families of harmonic functions in a planar domain with a small hole | |
Rivista: | JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS | |
Digital Object Identifier (DOI): | http://dx.doi.org/10.1016/j.jmaa.2014.08.037 | |
Volume: | 422 | |
Appare nelle tipologie: | 2.1 Articolo su rivista |
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140728_negative2D_els.pdf | preprint | Documento in Pre-print | Accesso chiuso-personale | Riservato |