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

2025/12/22 by Simone Creo, Creo, Simone, Salvatore Fragapane +1
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Mathematical Dynamics and Fractals #Advanced Mathematical Modeling in Engineering

paper · doi:10.48550/arxiv.2512.19252

Abstract

We study obstacle problems for the regional fractional p-Laplacian in a domain Ω⊂ℝ² 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⊂ℝ² having as boundary the n-th pre-fractal approximation of the Koch snowflake, for n∈ℕ. After proving the well-posedness of the approximating obstacle problems, we perform the asymptotic analysis for both n→+∞ and p→+∞.

Citations

• Convergence results for the solutions of (p,q)-Laplacian double obstacle problems on irregular domains
• Dynamic boundary conditions for time dependent fractional operators on extension domains
• Fine boundary regularity for the degenerate fractional p-Laplacian
• Obstacle problems and free boundaries: an overview
• Global regularity for the free boundary in the obstacle problem for the fractional Laplacian
• The abstract Lewy-Stampacchia inequality and applications
• Hitchhiker's guide to the fractional Sobolev spaces
• Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian

Related

• Convergence results for the solutions of (p,q)-Laplacian double obstacle problems on irregular domains
• Structure and regularity of the singular set in the obstacle problem for the fractional Laplacian
• The obstacle problem for the fractional Laplacian with critical drift
• Global regularity for the free boundary in the obstacle problem for the fractional Laplacian
• Variational inequalities for the fractional Laplacian