∞-Laplacian Obstacle Problems in Fractal Domains

Asymptotics for quasilinear obstacle problems in bad domains

“…The purpose of this paper is to give a complete answer to the asymptotic behavior of the solutions to the problems considered. Indeed, after the analysis of the behavior with respect to p, we will study the one with respect to n, showing that analogous results to the ones stated in [11] and [20] hold even in the case here examined. We stress the fact that in the proof of the convergence as n → ∞ a necessary step 2…”

“…In this section we focus our attention on the asymptotic behavior of the solutions. In particular, we prove an analogous result to the one presented in [6] for the homogeneous case without obstacles (see [4], [11], [20], [26] and [41] for the p-Laplacian case).…”

“…Then, it makes sense to ask whether the solutions to the approximating problems converge in some sense to a solution of the corresponding problem on Ω α . As far as we know, this double study on convergence, that is the analysis of the behavior with respect to n as well as on p, was done for first in [11], and then in [20]. Nevertheless, the study of the asymptotic behavior with respect to n was done by many authors for different problems (see, f.i., [9], [13], [14] and [42]).…”

“…A first step in this direction is done in [15], where the authors give sufficient conditions which allow to obtain the convergence of the whole sequence. Then, in [20] these conditions are used to state uniqueness results (in the case p = ∞) for the same unilateral obstacle problem.…”

“…In [20], uniqueness results for p-Laplacian unilateral problems are stated (see also [2], [4], [33] and the references quoted there for the problem of the uniqueness).…”