Singular perturbations of an elliptic operator with discontinuous nonlinearity

L.S. Frank, E.W. van Groesen

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Abstract

A family of nonlinear elliptic boundary value problems, affected by the presence of two positive parameters λ and ε, is considered. These problems arise in the theoretical treatment of membranes with enzymic activity, ε being in several realistic situations a small parameter compared to λ. We give an appropriate formulation of the reduced problem which corresponds to ε = 0. On the other hand there exists a critical value λC of λ such that for λ ≤ λC the problems with ε > 0 are regular and for λ > λC singular perturbations of the reduced problem. Some estimates for λC as a functional of the boundary function ϕ are derived and an explicit formula for the first variation of λC with respect to ϕ is established. Newton-Kantorovich iterative procedure is applied in order to approximate the solutions and to derive some asymptotic formulae. This procedure gives rise to an interesting class of linear coercive singular perturbations with discontinuous coefficients.

Original languageEnglish
Pages (from-to)289-303
Number of pages15
JournalNorth-Holland Mathematics Studies
Volume47
Issue numberC
DOIs
Publication statusPublished - 1 Jan 1981
Externally publishedYes

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