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

Author： Kavian Otared   Vogelius Michael  

Publisher： Royal Society of Edinburgh

ISSN： 1473-7124

Source： Proceedings Section A: Mathematics - Royal Society of Edinburgh, Vol.133, Iss.1, 2003-02, pp. : 119-149

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Abstract

Let Ω be a bounded C2,α domain in R2. We prove that the boundary-value problem Δv = 0 in Ω, ∂v/∂n = λ sinh(v) 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.

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