Analysis of Natural Function Spaces and Dynamics on Noncompact Manifolds under Symmetry ; Analysis natürlicher Funktionenräume und Dynamik auf nichtkompakten Mannigfaltigkeiten unter Symmetrie
Titel:
Analysis of Natural Function Spaces and Dynamics on Noncompact Manifolds under Symmetry ; Analysis natürlicher Funktionenräume und Dynamik auf nichtkompakten Mannigfaltigkeiten unter Symmetrie
Beteiligte:
Merker, Jochen
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Erschienen:
Staats- und Universitätsbibliothek Hamburg Carl von Ossietzky, 2005-01-01
Anmerkungen:
Diese Datenquelle enthält auch Bestandsnachweise, die nicht zu einem Volltext führen.
Beschreibung:
The main goal of this doctoral thesis is to discuss the foundations of dynamical systems, whose state space is a space of maps defined on a noncompact domain and whose dynamics are compatible with the symmetries of this domain. Obviously, a mathematical rigorous treatment of such dynamical systems requires to specify, which spaces of maps are used, e.g. Sobolev spaces. However, regarding pattern formation on noncompact manifolds under symmetry, solutions of dynamical equations within the class of Sobolev vector fields do not include typical patterns with noncompact symmetries, as they vanish at infinity. But even if solutions within other Banach spaces of maps can be established, the problem remains that for noncompact manifolds the symmetry group in general does not act continuously on Banach spaces of maps, as it acts by composition, but composition and evaluation are not continuous. Therefore, in this thesis locally convex spaces of maps like the local Sobolev spaces are used to model dynamical systems, where composition, evaluation and thus also the symmetry action are continuous. However, as the analysis of such natural function spaces is not far developed in literature, a main task of this thesis is to extend the analysis to such spaces, and to provide theorems used in the study of dynamical equations. Contrary to the category of normable spaces, the category of locally convex spaces is not tensorial closed, and thus there is no natural space of continuous linear maps between locally convex spaces. A problem is that it is not clear how to define continuously differentiable maps. However, by using a tensorial closed category of vector spaces endowed with a slightly more general topological structure than a locally convex topology, this problem can be solved and a sufficient differential calculus can be developed. But analysis requires more than just a differential calculus: Differential equations must be solved, an inverse function theorem is needed, and other theorems of classical analysis must be ...