On the relation between asymptotic behaviour of Neumann eigenvalue counting functions and the inner Minkowski content of fractal drums

Schmidt, Lucas Valentin (2024). On the relation between asymptotic behaviour of Neumann eigenvalue counting functions and the inner Minkowski content of fractal drums. University of Birmingham. Ph.D.

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Abstract

We define a class of well-foliated domains for which explicit lower bounds for the first non-trivial eigenvalue of the Neumann Laplacian are obtained. We show that $p$-Rohde snowflakes for $p \in \left[\frac{1}{4},\frac{\sqrt{3}-1}{2}\right)$ fall into this class. We establish conditions on snowflake-like domains and quasidisks that ensure well-foliatedness via bi-Lipschitz invariance. We prove that domains for which any sufficiently small inner $\epsilon$-parallel neighbourhood can be covered (with uniformly bounded multiplicity of the cover) by at most $C \epsilon^{-\delta}$ well-foliated domains allow an upper bound of the remainder term of the spectral counting function of the Neumann Laplacian at $t$ asymptotic to $t^{\delta/2}$. It is discussed how the cardinality of this cover relates to the upper inner Minkowski content and in particular we show that $p$-Rohde snowflakes satisfy this property whenever their upper inner Minkowski content is positive and finite. Applying the above results we obtain explicit bounds in the cases of homogeneous $p$-Rohde snowflakes for $p \in \left[\frac{1}{4},\frac{\sqrt{3}-1}{2}\right)$ and of the classical Koch snowflake. Finally we construct a family of fractal sprays whose gaps have fractal boundary and find non-trivial terms in the asymptotic expansion of the spectral counting function and the volume of the inner $\epsilon$-parallel neighbourhood with comparable scaling behaviour.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

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