TY - JOUR

T1 - Small perturbations in the type of boundary conditions for an elliptic operator

AU - Bonnetier, E.

AU - Dapogny, Charles

AU - Vogelius, Michael S.

N1 - Funding Information:
The authors are grateful to Jean-Claude Nédélec, who pointed out several useful references concerning Sections 5 and 6 of this work. EB is partially supported by the project ANR-17-CE40-0029 Multi-Onde, financed by the French Agence Nationale de la Recherche (ANR). The work of CD is partially supported by the project ANR-18-CE40-0013 SHAPO, financed by the French Agence Nationale de la Recherche (ANR). This work was carried out while MSV was on sabbatical at the University of Copenhagen and the Danish Technical University. This visit was in part supported by the Nordea Foundation and the Otto Mønsted Foundation. The work of MSV was also partially supported by NSF grant DMS-12-11330.
Funding Information:
The authors are grateful to Jean-Claude Nédélec, who pointed out several useful references concerning Sections 5 and 6 of this work. EB is partially supported by the project ANR-17-CE40-0029 Multi-Onde , financed by the French Agence Nationale de la Recherche (ANR). The work of CD is partially supported by the project ANR-18-CE40-0013 SHAPO , financed by the French Agence Nationale de la Recherche (ANR). This work was carried out while MSV was on sabbatical at the University of Copenhagen and the Danish Technical University. This visit was in part supported by the Nordea Foundation and the Otto Mønsted Foundation . The work of MSV was also partially supported by NSF grant DMS-12-11330 .
Publisher Copyright:
© 2022 Elsevier Masson SAS

PY - 2022/11

Y1 - 2022/11

N2 - In this article, we study the impact of a change in the type of boundary conditions of an elliptic boundary value problem. In the context of the conductivity equation we consider a reference problem with mixed homogeneous Dirichlet and Neumann boundary conditions. Two different perturbed versions of this “background” situation are investigated, when (i) The homogeneous Neumann boundary condition is replaced by a homogeneous Dirichlet boundary condition on a “small” subset ωε of the Neumann boundary; and when (ii) The homogeneous Dirichlet boundary condition is replaced by a homogeneous Neumann boundary condition on a “small” subset ωε of the Dirichlet boundary. The relevant quantity that measures the “smallness” of the subset ωε differs in the two cases: while it is the harmonic capacity of ωε in the former case, we introduce a notion of “Neumann capacity” to handle the latter. In the first part of this work we derive representation formulas that catch the structure of the first non trivial term in the asymptotic expansion of the voltage potential, for a general ωε, under the sole assumption that it is “small” in the appropriate sense. In the second part, we explicitly calculate the first non trivial term in the asymptotic expansion of the voltage potential, in the particular geometric situation where the subset ωε is a vanishing surfacic ball.

AB - In this article, we study the impact of a change in the type of boundary conditions of an elliptic boundary value problem. In the context of the conductivity equation we consider a reference problem with mixed homogeneous Dirichlet and Neumann boundary conditions. Two different perturbed versions of this “background” situation are investigated, when (i) The homogeneous Neumann boundary condition is replaced by a homogeneous Dirichlet boundary condition on a “small” subset ωε of the Neumann boundary; and when (ii) The homogeneous Dirichlet boundary condition is replaced by a homogeneous Neumann boundary condition on a “small” subset ωε of the Dirichlet boundary. The relevant quantity that measures the “smallness” of the subset ωε differs in the two cases: while it is the harmonic capacity of ωε in the former case, we introduce a notion of “Neumann capacity” to handle the latter. In the first part of this work we derive representation formulas that catch the structure of the first non trivial term in the asymptotic expansion of the voltage potential, for a general ωε, under the sole assumption that it is “small” in the appropriate sense. In the second part, we explicitly calculate the first non trivial term in the asymptotic expansion of the voltage potential, in the particular geometric situation where the subset ωε is a vanishing surfacic ball.

KW - Asymptotic analysis

KW - Integral representation

KW - Logarithmic capacity

KW - Neumann capacity

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U2 - 10.1016/j.matpur.2022.09.003

DO - 10.1016/j.matpur.2022.09.003

M3 - Article

AN - SCOPUS:85139347581

SN - 0021-7824

VL - 167

SP - 101

EP - 174

JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

ER -