  In this paper we study the equilibrium shape of an interface that represents
the lateral boundary of a pore channel embedded in an elastomer. The model
consists of a system of PDEs, comprising a linear elasticity equation for
displacements within the elastomer and a nonlinear Poisson equation for the
electric potential within the channel (filled with protons and water). To
determine the equilibrium interface, a variational approach is employed. We
analyze: i) the existence and uniqueness of the electrical potential, ii) the
shape derivatives of state variables and iii) the shape differentiability of
the corresponding energy and the corresponding Euler-Lagrange equation. The
latter leads to a modified Young-Laplace equation on the interface. This
modified equation is compared with the classical Young-Laplace equation by
computing several equilibrium shapes, using a fixed point algorithm.
