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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Vydáno v:Mathematical methods in the applied sciences Ročník 41; číslo 13; s. 5211 - 5229
Hlavní autoři: Musolino, Paolo, Mishuris, Gennady
Médium: Journal Article
Jazyk:angličtina
Vydáno: Freiburg Wiley Subscription Services, Inc 15.09.2018
Témata:
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
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Shrnutí: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.
Bibliografie:ObjectType-Article-1
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content type line 14
ISSN:0170-4214
1099-1476
DOI:10.1002/mma.5072