  In 1993, V. \v{S}ver\'ak proved that if a sequence of uniformly bounded
domains $\Omega_n\subset {\mathbb R}^2$ such that $\Omega_n\to \Omega$ in the
sense of the Hausdorff complementary topology, verify that the number of
connected components of its complements are bounded, then the solutions of the
Dirichlet problem for the Laplacian with source $f\in L^2({\mathbb R}^2)$
converges to the solution of the limit domain with same source. In this paper,
we extend \v{S}ver\'ak result to variable exponent spaces.
