A nonlinear problem for the Laplace equation with a degenerating Robin condition

Paolo Musolino, Gennady Mishuris

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Abstract

We investigate the behavior of the solutions of a mixed problem for the Laplace equation in a domain Ω. On a part of the boundary ∂Ω, we consider a Neumann condition, whereas in another part, we consider a nonlinear Robin condition, which depends on a positive parameter δ in such a way that for δ = 0 it degenerates into a Neumann condition. For δ small and positive, we prove that the boundary value problem has a solution u(δ,·). We describe what happens to u(δ,·) as δ→0 by means of representation formulas in terms of real analytic maps. Then, we confine ourselves to the linear case, and we compute explicitly the power series expansion of the solution
Original languageEnglish
Pages (from-to)5211-5229
Number of pages19
JournalMathematical Methods in the Applied Sciences
Volume41
Issue number13
Early online date16 May 2018
DOIs
Publication statusPublished - 15 Sept 2018

Keywords

  • boundary value problems for second-order elliptic equations
  • integral equations methods
  • Laplace operator
  • Neumann problem
  • Robin problem
  • singularly perturbed problem

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