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

TL;DR

This paper investigates obstacle problems involving the fractional p-Laplacian on fractal domains with Koch snowflake boundaries, establishing well-posedness, equivalent formulations, and asymptotic behaviors as domain complexity and p grow large.

Contribution

It introduces the first well-posedness results for obstacle problems on fractal domains with fractional operators and analyzes their asymptotic limits.

Findings

Well-posedness of obstacle problems on Koch snowflake domains.

Equivalent formulations of the obstacle problem.

Asymptotic behavior as domain complexity and p tend to infinity.

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 $p\to+\infty$.