eprintid: 5580
rev_number: 12
eprint_status: archive
userid: 1
dir: disk0/00/00/55/80
datestamp: 2005-06-09 13:55:17
lastmod: 2015-05-02 05:15:12
status_changed: 2012-08-14 15:15:09
type: preprint
metadata_visibility: show
creators_name: Lorenz, Thomas
title: Generalizing evolution equations in ostensible metric spaces: Timed right-hand sleek solutions provide uniqueness of first-order geometric examples.
ispublished: pub
subjects: ddc-510
divisions: i-708000
keywords: generalized ODE on sets with nonsymmetric distance functions , nonlocal geometric evolutions , mutational equations , set-valued maps
cterms_swd: Verallgemeinerte Differentialgleichung
cterms_swd: Nichtlineare Evolutionsgleichung
cterms_swd: Asymmetrische Metrik
cterms_swd: Mengenwertige Abbildung
cterms_swd: Nichtglatte Analysis
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. 
abstract_translated_lang: eng
class_scheme: msc
class_labels: 49J52, 54E99, 34G20, 34A60
date: 2005
date_type: published
id_scheme: DOI
id_number: 10.11588/heidok.00005580
portal_cluster_id: p-iwrpp
portal_order: 05580
ppn_swb: 1646175379
own_urn: urn:nbn:de:bsz:16-heidok-55802
language: eng
bibsort: LORENZTHOMGENERALIZI2005
full_text_status: public
series: IWR-Preprints
citation:   Lorenz, Thomas  (2005) Generalizing evolution equations in ostensible metric spaces: Timed right-hand sleek solutions provide uniqueness of first-order geometric examples.  [Preprint]     
document_url: https://archiv.ub.uni-heidelberg.de/volltextserver/5580/1/Lorenz_sleek_solutions.pdf.old
document_url: https://archiv.ub.uni-heidelberg.de/volltextserver/5580/2/Lorenz_sleek_solutions_SVA.pdf