Regularity and asymptotics for p-Laplace type operators in fractal and pre-fractal domains

In this thesis we deal with double obstacle problems involving p-Laplace type operators in fractal and pre-fractal domains of R^2. The first improvement here presented consists in establishing a regularity result for the solution to double obstacle problem in terms of weighted Sobolev spaces (involving second derivatives), where we take as weight the distance from the vertex of the reentrant corner. In particular, we prove local estimates, estimates far away from the conical point and the boundedness of the gradient far away from the conical point. In addition, thanks to the regularity result presented, we establish a sharp error estimates for the FEM approximations follow the approach of P. Grisvard, considering a suitable triangulation of the domains, adapted to the regularity of the solutions. Furthermore, after stating this optimal estimate, we perform numerical simulations investigating both the cases of p = 2 and p > 2 (fixed). In the last part of the work, considering both the case of fractal domains and the n-th pre-fractal ones, we investigate the asymptotic behaviour of the solution with respect to p and n. Moreover, we briefly discuss the issue of the uniqueness.