Arzt, Peter: Eigenvalues of measure theoretic Laplacians on Cantor-like setsEigenwerte von maßtheoretischen Laplace-Operatoren auf Cantor-artigen Mengen. 2014
Inhalt
- Abstract / Zusammenfassung
- Contents
- 1. Introduction
- 1.1. Statement of the problem
- 1.2. Physical motivation for the Laplacian
- 1.3. The Cantor set and generalizations
- 1.4. Outline of the thesis
- 2. Preliminaries
- 2.1. Derivatives and the Laplacian with respect to ameasure
- 2.2. Dirichlet forms related to the Laplacian
- 2.3. A Poincaré inequality
- 3. Spectral Asymptotics for General Homogeneous Cantor Measures
- 3.1. Construction of general homogeneous Cantormeasures
- 3.2. Scaling of the eigenvalue counting functions
- 3.3. Spectral asymptotics
- 3.4. Deterministic examples
- 3.5. Application to random homogeneous measures
- 4. Eigenvalues of the Laplacian as Zeros of Generalized Sine Functions
- 4.1. Generalized trigonometric functions
- 4.2. Calculation of L2-norms
- 4.3. A trigonometric identity
- 4.4. Symmetric measures
- 4.5. Self-similar measures
- 4.6. Self-similar measures with r1m1=r2m2
- 4.7. Self-similar measures with r1m1=r2m2 and r1+r2=1
- 4.8. Figures and numbers
- 4.9. Remarks and outlook
- A. Plots of Eigenfunctions
- B. Mathematical Foundations
- B.1. L2 spaces
- B.2. Self-similar sets and measures
- B.3. The Vitali-Hahn-Saks theorem
- B.4. The Arzelà-Ascoli theorem
- B.5. The law of the iterated logarithm
- B.6. Regularly varying functions
- B.7. Dirichlet forms
- Bibliography