velikost textu

Properties of mappings of finite distortion

Upozornění: Informace získané z popisných dat či souborů uložených v Repozitáři závěrečných prací nemohou být použity k výdělečným účelům nebo vydávány za studijní, vědeckou nebo jinou tvůrčí činnost jiné osoby než autora.
Název:
Properties of mappings of finite distortion
Název v češtině:
Vlastnosti zobrazení s konečnou distorzí
Typ:
Disertační práce
Autor:
Mgr. Daniel Campbell
Školitel:
doc. RNDr. Stanislav Hencl, Ph.D.
Oponenti:
Prof. Pekka Koskela
Dr. Carlos Mora Corral
Id práce:
109733
Fakulta:
Matematicko-fyzikální fakulta (MFF)
Pracoviště:
Katedra matematické analýzy (32-KMA)
Program studia:
Matematika (P1101)
Obor studia:
Matematická analýza (4M3)
Přidělovaný titul:
Ph.D.
Datum obhajoby:
23. 6. 2017
Výsledek obhajoby:
Prospěl/a
Jazyk práce:
Angličtina
Klíčová slova:
Zobrazení s konečnou distorzí, otevřenost a diskrétnost
Klíčová slova v angličtině:
Mapping of finite distortion, openess and disreteness
Abstrakt:
Abstract: In the following thesis we will be mostly concerned with questions related to the regularity of solutions to non-linear elasticity models in the calculus of variations. An important step in this is question is the approximation of Sobolev homeomorphisms by diffeomorphisms. We refine an approximation result of Hencl and Pratelli’s which, for a given planar Sobolev (or Sobolev-Orlicz) homeomorphism, constructs a diffeomorphism arbitrarily close to the original map in uniform convergence and in terms of the Sobolev-Orlicz norm. Further we show, in dimension 4 or higher, that such an approximation result cannot hold in Sobolev spaces W 1,p where p is too small by constructing a sense-preserving homeomorphism with Jacobian negative on a set of positive measure. The class of mappings referred to as mappings of finite distortion have been proposed as possible models for deformations of bodies in non-linear elasticity. In this context a key property is their continuity. We show, by counter-example, the surprising sharpness of the modulus of continuity with respect to the integrability of the distortion function. Also we prove an optimal regularity result for the inverse of a bi-Lipschitz Sobolev map in W k,p and composition of Lipschitz maps in W k,p comparable with the classical inverse mapping theorem. As a consequence we retrieve a Sobolev equivalent of the implicit function theorem. 1
Abstract v angličtině:
Abstract: In the following thesis we will be mostly concerned with questions related to the regularity of solutions to non-linear elasticity models in the calculus of variations. An important step in this is question is the approximation of Sobolev homeomorphisms by diffeomorphisms. We refine an approximation result of Hencl and Pratelli’s which, for a given planar Sobolev (or Sobolev-Orlicz) homeomorphism, constructs a diffeomorphism arbitrarily close to the original map in uniform convergence and in terms of the Sobolev-Orlicz norm. Further we show, in dimension 4 or higher, that such an approximation result cannot hold in Sobolev spaces W 1,p where p is too small by constructing a sense-preserving homeomorphism with Jacobian negative on a set of positive measure. The class of mappings referred to as mappings of finite distortion have been proposed as possible models for deformations of bodies in non-linear elasticity. In this context a key property is their continuity. We show, by counter-example, the surprising sharpness of the modulus of continuity with respect to the integrability of the distortion function. Also we prove an optimal regularity result for the inverse of a bi-Lipschitz Sobolev map in W k,p and composition of Lipschitz maps in W k,p comparable with the classical inverse mapping theorem. As a consequence we retrieve a Sobolev equivalent of the implicit function theorem. 1
Dokumenty
Stáhnout Dokument Autor Typ Velikost
Stáhnout Text práce Mgr. Daniel Campbell 2.97 MB
Stáhnout Abstrakt v českém jazyce Mgr. Daniel Campbell 41 kB
Stáhnout Abstrakt anglicky Mgr. Daniel Campbell 41 kB
Stáhnout Posudek vedoucího doc. RNDr. Stanislav Hencl, Ph.D. 45 kB
Stáhnout Posudek oponenta Prof. Pekka Koskela 1.08 MB
Stáhnout Posudek oponenta Dr. Carlos Mora Corral 215 kB
Stáhnout Záznam o průběhu obhajoby prof. RNDr. Luboš Pick, CSc., DSc. 104 kB