Lecture Notes on Mean Curvature Flow: Barr...

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Sprache: Englisch

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Beschreibung

The aim of the book is to study some aspects of geometric evolutions, such as mean curvature flow and anisotropic mean curvature flow of hypersurfaces. We analyze the origin of such flows and their geometric and variational nature. Some of the most important aspects of mean curvature flow are described, such as the comparison principle and its use in the definition of suitable weak solutions. The anisotropic evolutions, which can be considered as a generalization of mean curvature flow, are studied from the view point of Finsler geometry. Concerning singular perturbations, we discuss the convergence of the Allen¿Cahn (or Ginsburg¿Landau) type equations to (possibly anisotropic) mean curvature flow before the onset of singularities in the limit problem. We study such kinds of asymptotic problems also in the static case, showing convergence to prescribed curvature-type problems.

Zusammenfassung

Elementary introduction to mean curvature flow, including the short time existence result, using the signed distance function

Detailed study of minimal barriers and their regularizations for mean curvature flow

An example of fattening

Convergence and error estimate for the parabolic Allen-Cahn equation

Inhaltsverzeichnis

Signed distance from a smooth boundary.- Mean curvature vector and second fundamental form.- First variations of volume integrals and of the perimeter.- Smooth mean curvature flows.- Huisken's monotonicity formula.- Inclusion principle. Local well posedness: the approach of Evans-Spruck.- Grayson's example.- De Giorgi's barriers.- Inner and outer regularizations.- An example of fattening.- Ilmanen's interposition lemma.- The avoidance principle.- Comparison between barriers and a generalized evolution.- Barriers and level set evolution.- Parabolic singular perturbations: formal matched asymptotics, convergence and error estimate.

Details

Erscheinungsjahr:	2014
Fachbereich:	Geometrie
Genre:	Importe , Mathematik
Rubrik:	Naturwissenschaften & Technik
Medium:	Taschenbuch
Reihe:	Publications of the Scuola Normale Superiore
Inhalt:	xviii 329 S.
ISBN-13:	9788876424281
ISBN-10:	8876424288
Sprache:	Englisch
Einband:	Kartoniert / Broschiert
Autor:	Bellettini, Giovanni
Hersteller:	Edizioni della Normale Publications of the Scuola Normale Superiore
Verantwortliche Person für die EU:	Springer Verlag GmbH, Tiergartenstr. 17, D-69121 Heidelberg, juergen.hartmann@springer.com
Maße:	235 x 155 x 19 mm
Von/Mit:	Giovanni Bellettini
Erscheinungsdatum:	16.01.2014
Gewicht:	0,528 kg