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On the existence and ‘blow-up’ of solutions to a two-dimensional nonlinear boundary-value problem arising in corrosion modelling

Published online by Cambridge University Press:  12 July 2007

Otared Kavian  and
Michael Vogelius
Otared Kavian
Affiliation:
Lab. de Math. Appliquées (UMR 7641), Université de Versailles, 45 avenue des États Unis, 78035 Versailles Cedex, France ([email protected])
Michael Vogelius
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA ([email protected])
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Abstract

Let Ω be a bounded C2,α domain in R2. We prove that the boundary-value problem Δυ = 0 in Ω, ∂υ/∂n = λsinh(υ) on ∂Ω, has infinitely many (classical) solutions for any given λ > 0. These solutions are constructed by means of a variational principle. We also investigate the limiting behaviour as λ → 0+; indeed, we prove that each of our solutions, as λ → 0+, after passing to a subsequence, develops a finite number of singularities located on ∂Ω.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 133 , Issue 1 , February 2003 , pp. 119 - 149
Copyright
Copyright © Royal Society of Edinburgh 2003

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