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

TL;DR

This paper develops obstacle problems for the regional fractional $p$-Laplacian on domains with Koch-type fractal boundaries, establishing well-posedness, equivalent formulations, and Robin-type boundary conditions. It then analyzes asymptotic behavior in two directions: as the nonlinearity exponent $p$ tends to infinity and as the fractal boundary is approximated by pre-fractal domains, including joint limits, proving convergence to limit maximization problems on fractal domains and corresponding pre-fractal problems. The results provide the first rigorous treatment of obstacle problems in fractal-domain settings for nonlocal quasi-linear operators, with precise convergence mechanisms and boundary-condition formulations, contributing to both theory and potential numerical applications. They also connect the obstacle problem to free boundary phenomena and offer pathways to extend to more general Koch-type fractals and irregular domains.

Abstract

We study obstacle problems for the regional fractional $p$-Laplacian in a domain $Ω\subset\mathbb{R}^2$ 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 $Ω_n\subset\mathbb{R}^2$ 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 $p\to+\infty$.

Figures (1)

• Figure 1: The Koch snowflake $K$.

Theorems & Definitions (21)

• Definition 1.1
• Definition 1.2
• Proposition 1.1
• Theorem 1.1
• Theorem 1.2
• Theorem 2.1: Fractional Green formula
• Proposition 2.1
• Lemma 2.1
• Lemma 2.2
• Theorem 2.2