  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.
