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Linear elliptic boundary value problems in varying domains
Author(s) -
Bochniak Marius
Publication year - 2003
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310019
Subject(s) - mathematics , boundary value problem , parametrix , perturbation (astronomy) , differentiable function , mathematical analysis , elliptic operator , elliptic boundary value problem , differential operator , poincaré–steklov operator , mixed boundary condition , semi elliptic operator , robin boundary condition , physics , quantum mechanics
Abstract The paper is devoted to the study of solutions to linear elliptic boundary value problems in domains depending smoothly on a small perturbation parameter. To this end we transform the boundary value problem onto a fixed reference domain and obtain a problem in a fixed domain but with differential operators depending on the perturbation parameter. Using the Fredholm property of the underlying operator we show the differentiability of the transformed solution under the assumption that the dimension of the kernel does not depend on the perturbation parameter. Furthermore, we obtain an explicit representation for the corresponding derivative.