porn math

So, if you want to prove regularity of the week solution for the eigenvalue problem for the Laplacian on a smooth domain Ω, you usually proceed like this. If your data is L²(Ω), then the solution is H²(Ω). Since, now, your solution is also your data, your data is in fact H²(Ω), so that your solution is more regular. You obtain smoothness in the sense of Sobolev if you bootstrap on and on and on...