In this thesis we study the asymptotic behavior of bi-Lipschitz diffeomorphic weighted Riemannian manifolds with techniques from the theory of homogenization. To do so we re-interpret the problem as different induced metrics on one reference manifold.
Our analysis is twofold. On the one hand we consider second-order uniformly elliptic operators on weighted Riemannian manifolds. They naturally emerge when studying spectral properties of the Laplace-Beltrami operator on families of manifolds with rapidly oscillating metrics. We appeal to the notion of H-convergence introduced by Murat and Tartar. In our first main result we establish an H-compactness result that applies to elliptic operators with measurable, uniformly elliptic coefficients on weighted Riemannian manifolds. We further discuss the special case of locally periodic coefficients and study the asymptotic spectral behavior of Euclidean submanifolds with rapidly oscillating geometry.
On the other hand we study integral functionals featuring non-convex integrands with non-standard growth on the Euclidean space in a stochastic framework. Our second main result is a Γ-convergence statement under certain assumptions on the statistics of their integrands. Such functionals provide a tool to study the Dirichlet energy on non-uniformly bi-Lipschitz diffeomorphic manifolds. We show Mosco-convergence of the Dirichlet energy and deduce conditions for the spectral behavior of weighted Riemannian manifolds with locally oscillating random structure, especially in the case of Euclidean submanifolds.:Introduction
Outline
Notation
I. Preliminaries
1. Convergence of Riemannian Manifolds
1.1. Hausdorff-Convergence
1.2. Gromov-Hausdorff-Convergence
1.3. Spectral Convergence
1.4. Mosco-Convergence
2. Homogenization
2.1. Periodic Homogenization
2.2. Stochastic Homogenization
II. Uniformly bi-Lipschitz Diffeomorphic Manifolds
3. Uniformly Elliptic Operators on a Riemannian Manifold
3.1. Setting
3.2. Main Results
3.3. Strategy of the Proof and Auxiliary Results
3.4. Identi cation of the Limit via Local Coordinate Charts
3.5. Examples
3.6. Proofs
4. Application to Uniformly bi-Lipschitz Diffeomorphic Manifolds
4.1. Setting and Results
4.2. Examples
4.3. Proofs
III. Rapidly Oscillating Random Manifolds
5. Integral Functionals with Non-Uniformal Growth
5.1. Setting
5.2. Main Results
5.3. Strategy of the Proof and Auxiliary Results
5.4. Proofs
6. Application to Rapidly Oscillating Riemannian Manifolds
6.1. Setting and Results
6.2. Examples
6.3. Proofs
Summary and Discussion
Bibliography
List of Figures
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:74376 |
| Date | 12 April 2021 |
| Creators | Hoppe, Helmer |
| Contributors | Neukamm, Stefan, Schlömerkemper, Anja, Technische Universität Dresden |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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