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

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 = ...

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Bibliographic Details
Published in:Mathematical methods in the applied sciences Vol. 41; no. 13; pp. 5211 - 5229
Main Authors: Musolino, Paolo, Mishuris, Gennady
Format: Journal Article
Language:English
Published: Freiburg Wiley Subscription Services, Inc 15.09.2018
Subjects:
Boundary value problems
boundary value problems for second‐order elliptic equations
integral equations methods
Laplace equation
Laplace operator
Neumann problem
Power series
Robin problem
Series expansion
singularly perturbed problem
ISSN:0170-4214, 1099-1476
Online Access:Get full text
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Summary: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.
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ISSN:0170-4214
1099-1476
DOI:10.1002/mma.5072