Asymptotic expansion of the voltage potential under small Robin perturbations of its boundary conditions
Abstract
Inspired by questions related to inverse problems and shape optimization, we derive an asymptotic expansion of the voltage potential, solution to a model elliptic second-order partial differential equation, under small perturbations of its boundary conditions. More precisely, the homogeneous Dirichlet or homogeneous Neumann boundary condition in a fixed, reference configuration of the problem is replaced by a Robin boundary condition with admittance kε > 0 on a "small" subset ωε of the boundary of the ambient domain, vanishing at the limit ε → 0. In each of these two situations, a general asymptotic formula is established for the voltage potential, which rests on minimal assumptions about the shape of the vanishing subset ωε and the parameter kε. The scalings of these expansions, capturing the intensity of the perturbation, are measured by new quantities called "Robin-Dirichlet" capacity or a "Robin-Neumann" capacity, which depend on the geometry of ωε and on the value of the parameter kε. We analyze how these quantities compare to more classical measures of the "smallness" of ωε, such as the capacity or the Neumann capacity of ωε, according to the behavior of kε as ε → 0.
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BibTeX
@misc{dapogny2026asymptotic,
title = {{Asymptotic expansion of the voltage potential under small Robin perturbations of its boundary conditions}},
author = {Charles Dapogny and Eric Bonnetier and Roman Moskalenko},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05740259v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Asymptotic expansion of the voltage potential under small Robin perturbations of its boundary conditions
AU - Charles Dapogny
AU - Eric Bonnetier
AU - Roman Moskalenko
PY - 2026
DA - 2026/09/07
PB - HAL
UR - https://hal.science/hal-05740259v1
AB - Inspired by questions related to inverse problems and shape optimization, we derive an asymptotic expansion of the voltage potential, solution to a model elliptic second-order partial differential equation, under small perturbations of its boundary conditions. More precisely, the homogeneous Dirichlet or homogeneous Neumann boundary condition in a fixed, reference configuration of the problem is replaced by a Robin boundary condition with admittance kε > 0 on a "small" subset ωε of the boundary of the ambient domain, vanishing at the limit ε → 0. In each of these two situations, a general asymptotic formula is established for the voltage potential, which rests on minimal assumptions about the shape of the vanishing subset ωε and the parameter kε. The scalings of these expansions, capturing the intensity of the perturbation, are measured by new quantities called "Robin-Dirichlet" capacity or a "Robin-Neumann" capacity, which depend on the geometry of ωε and on the value of the parameter kε. We analyze how these quantities compare to more classical measures of the "smallness" of ωε, such as the capacity or the Neumann capacity of ωε, according to the behavior of kε as ε → 0.
ER -