A duality approach or the boundary variation of Neumann problems

In two dimensions, we study the stability of the solution of an elliptic equation with Neumann boundary conditions for nonsmooth perturbations of the geometric domain. Using harmonic conjugates, we relate this problem to the shape stability of the solution of an elliptic equation with Dirichlet boundary conditions. As a particular case, we prove the stability of the solution under a topological constraint ( uniform number of holes), which is analogous to Sverak's result for Dirichlet boundary conditions.