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Generalizing evolution equations in ostensible metric spaces: Timed right-hand sleek solutions provide uniqueness of first-order geometric examples.

Lorenz, Thomas

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Abstract

The mutational equations of Aubin extend ordinary differential equations to metric spaces (with compact balls). In first-order geometric evolutions, however, the topological boundary need not be continuous in the sense of Painleve–Kuratowski. So this paper suggests a generalization of Aubin’s mutational equations that extends classical notions of dynamical systems and functional analysis beyond the traditional border of vector spaces: Distribution– like solutions are introduced in a set just supplied with a countable family of (possibly non-symmetric) distance functions. Moreover their existence is proved by means of Euler approximations and a form of “weak” sequential compactness (although no continuous linear forms are available beyond topological vector spaces). This general framework is applied to a first-order geometric example, i.e. compact subsets of the Euclidean space evolving according to the nonlocal properties of both the current set and its proximal normal cones. Here neither regularity assumptions about the boundaries nor the inclusion principle are required. In particular, we specify sufficient conditions for the uniqueness of these solutions.

Item Type: Preprint
Series Name: IWR-Preprints
Date Deposited: 09 Jun 2005 13:55
Date: 2005
Faculties / Institutes: Service facilities > Interdisciplinary Center for Scientific Computing
Subjects: 510 Mathematics
Controlled Keywords: Verallgemeinerte Differentialgleichung, Nichtlineare Evolutionsgleichung, Asymmetrische Metrik, Mengenwertige Abbildung, Nichtglatte Analysis
Uncontrolled Keywords: generalized ODE on sets with nonsymmetric distance functions , nonlocal geometric evolutions , mutational equations , set-valued maps
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