Eigenvalue counting functions and parallel volumes for examples of fractal sprays generated by the Koch snowflake

Abstract

We apply recent results by the authors to obtain bounds on remainder terms of the Dirichlet Laplace eigenvalue counting function for domains that can be realised as countable disjoint unions of scaled Koch snowflakes. Moreover we compare the resulting exponents to the exponents in the asymptotic expansion of the domain's inner parallel volume.

Figures (6)

• Figure 1: Depiction of the IFS of the fractal spray studied in Sec. \ref{['sec:weylasymp']} and Sec. \ref{['sec:minkcontent']}. The base length$b$ of the snowflake $K$ is also shown.
• Figure 2: Depiction of the variant $\Omega(1,0)$ of the IFS of the fractal spray studied in Sec. \ref{['sec:weylasymp']} and Sec. \ref{['sec:minkcontent']}. In this case, the map $\phi_1$ has been replaced by the six maps $\phi_{1,1},\ldots,\phi_{1,6}$ giving rise to an additional connected component of the generator $G=K_0 \cup K_1$. Analogously one can replace $\phi_7,\ldots,\phi_{12}$.
• Figure 3: Example of the iterative construction of $\Omega$ as defined in \ref{['eq:Omega']}. From left to right the $0^{\text{th}}$, first and second iterations of the generator $K$ under $\Phi$ are shown, rotated by $30^\circ$. The $0^{\text{th}}$ iteration (left) shows $K$. The first iteration (middle) shows $K \cup \Phi(K)$ . The second iteration (right) then shows $K \cup \Phi(K) \cup \Phi^2(K)$.
• Figure 4: Instance of a volume covering domain as used in the proof of Thm. \ref{['thm:satz1']}.
• Figure :

Theorems & Definitions (12)

• Proposition 3.1: Dirichlet-Neumann bracketing with multiplicity, NetrusovSafarov2005
• Theorem 3.2: cf. DOKUMENT
• proof : Sketch of proof
• Remark 3.3
• Theorem 4.1
• proof
• Remark 4.2
• Theorem 5.1
• Lemma 5.2
• Remark 5.3