$p(x)$-Stability of the Dirichlet problem for Poisson's equation with variable exponents

It is shown that if the sequence $(p_j(x))$ increases uniformly to $p(x)$ in a bounded, smooth domain $\Omega$, then the sequence $(u_i)$ of solutions to the Dirichlet problem for the $p_i(x)$-Laplacian with fixed boundary datum $\varphi$ converges (in a sense to be made precise) to the solution $u_p$ of the Dirichlet problem for the $p(x)$-Laplacian with boundary datum $\varphi$. A similar result is proved for a decreasing sequence $p_j\searrow p$