Eigenfunctions of the Koch snowflake domain

Abstract

We obtain non-tangential boundary estimates for the Dirichlet eigenfunctions ϕn and their gradients {∇ϕn} for a class of planar domains Ω with fractal boundaries. This class includes the quasidiscs and, in particular, snowflake-type domains. When applied to the case when Ω is the Koch snowflake domain, one of our main results states that {∇ϕ1(ω)} tends to ∞ or 0 as ω approaches certain types of boundary points (where ϕ1 > 0 denotes the ground state eigenfunction of the Dirichlet Laplacian on Ω). More precisely, let Ob (resp., Ac) denote the set of boundary points which are vertices of obtuse (resp., acute) angles in an inner polygonal approximation of the snowflake curve ∂Ω. Then given νeOb (resp., νe Ac), we show that {∇ϕ1(ω)}→∞ (resp., 0) as ω tends to ν in Φ within a cone based at ν. Moreover, we show that blowup of {∇ϕ1} also occurs at all boundary points in a Cantor-type set. These results have physical relevance to the damping of waves by fractal coastlines, as pointed out by Sapovalet al. in their experiments on the “Koch drum”.