Continuity of neumann linear elliptic problems on varying two—dimensional bounded open sets

Given a sequence of uniformly bounded open sets of the plane, whose boundaries are connected and uniformly bounded in length, converging to some open set (in the sense that the complements converge to for the Hausdorffmetric), we show that the Neumann solutions ui of (where with all i) converge strongly in L2(B) to the solution of the same problem on . We also get the strong convergence of the gradients. From this we deduce that, given any , there exists a sequence of functions that converges strongly to u, and such that converges strongly to .