Obstacle problems for the fractional $p$-Laplacian on fractal domains: well-posedness and asymptotics

Abstract

We study obstacle problems for the regional fractional $p$-Laplacian in a domain $\Omega\subset\mathbb{R}^2$ having as fractal boundary the Koch snowflake. We prove well-posedness results for the solution of the obstacle problem, as well as two equivalent formulations. Moreover, we study corresponding approximating obstacle problems in a sequence of domains $\Omega_n\subset\mathbb{R}^2$ having as boundary the $n$-th pre-fractal approximation of the Koch snowflake, for $n\in\mathbb{N}$. After proving the well-posedness of the approximating obstacle problems, we perform the asymptotic analysis for both $n\to+\infty$ and $p\to+\infty$.