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

TL;DR

This paper investigates how small changes in boundary conditions of an elliptic boundary value problem affect the solution, introducing new capacity concepts and deriving asymptotic formulas for voltage potential perturbations.

Contribution

It introduces a new notion of Neumann capacity and provides explicit asymptotic formulas for boundary condition perturbations in elliptic problems.

Findings

Derived representation formulas for the first non-trivial term in asymptotic expansion.

Explicit calculation of the asymptotic term for a surfacic ball perturbation.

Established the relevance of harmonic and Neumann capacities in boundary perturbation analysis.

Abstract

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 $\omega_\varepsilon$ of the Neumann boundary; and when (ii) The homogeneous Dirichlet boundary condition is replaced by a homogeneous Neumann boundary condition on a ``small'' subset $\omega_\varepsilon $ of the Dirichlet boundary. The relevant quantity that measures the ``smallness'' of the subset $\omega_\varepsilon $ differs in the two cases: while it is the harmonic capacity of $\omega_\varepsilon $ 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 $\omega_\varepsilon $, 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 $\omega_\varepsilon $ is a vanishing surfacic ball.