Asymptotic expansion of the voltage potential under small Robin perturbations of its boundary conditions
Year: 2026
Type: preprint
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 $k_\varepsilon > 0$ on a ``small'' subset $ω_\varepsilon$ of the boundary of the ambient domain, vanishing at the limit $\varepsilon \to 0$. 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 $ω_\varepsilon$ and the parameter $k_\varepsilon$. 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 $ω_\varepsilon$ and on the value of the parameter $k_\varepsilon$. We analyze how these quantities compare to more classical measures of the ``smallness'' of $ω_\varepsilon$, such as the capacity or the Neumann capacity of $ω_\varepsilon$, according to the behavior of $k_\varepsilon$ as $\varepsilon \to 0$.
Source: arXiv (Cornell University)
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7211910795