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

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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.