Continuity of Neumann linear elliptic problems on varying two-dimensional bounded open sets.

Given a sequence (Omega(i))(i greater than or equal to 1) of uniformly bounded open sets of the plane, whose boundaries delta Omega(i) are connected and uniformly bounded in length, converging to some open set Omega (in the sense that the complements Omega(i)(c) converge to Omega(c) for the Hausdorff metric), we show that the Neumann solutions u(i) of -Delta u + lambda u = v in Omega(i) (where v is an element of L-2(B) with B superset of Omega; for all i) converge strongly in L-2(B) to the solution of the same problem on Omega. We also get the strong convergence of the gradients. From this we deduce that, given any u is an element of H-1(Omega), there exists a sequence of functions u(i) is an element of H-1(Omega(i)) that converges strongly to u, and such that (del u(i))(i greater than or equal to 1) converges strongly to del u.