Continuity of neumann linear elliptic problems on varying two—dimensional bounded open sets
Antonin Chambolle, F Doveri · Communications in Partial Differential Equations · 1997
Given a sequenceof uniformly bounded open sets of the plane, whose boundariesare connected and uniformly bounded in length, converging to some open set(in the sense that the complementsconverge to for the Hausdorffmetric), we show that the Neumann solutions ui of(wherewithall 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 functionsthat converges strongly to u, and such thatconverges strongly to.