
Properties of mappings of finite distortion
Campbell, Daniel ; Hencl, Stanislav (advisor) ; Koskela, Pekka (referee) ; Mora Corral, Carlos (referee)
In the following thesis we will be mostly concerned with questions related to the regularity of solutions to nonlinear 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 SobolevOrlicz) homeomorphism, constructs a diffeomorphism arbitrarily close to the original map in uniform convergence and in terms of the SobolevOrlicz norm. Further we show, in dimension 4 or higher, that such an approximation result cannot hold in Sobolev spaces W1,p where p is too small by constructing a sensepreserving 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 nonlinear elasticity. In this context a key property is their continuity. We show, by counterexample, 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 biLipschitz Sobolev map in Wk,p and composition of Lipschitz maps in Wk,p comparable with the classical inverse mapping theorem. As a...


Properties of mappings of finite distortion
Campbell, Daniel ; Hencl, Stanislav (advisor)
In the following thesis we will be mostly concerned with questions related to the regularity of solutions to nonlinear 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 SobolevOrlicz) homeomorphism, constructs a diffeomorphism arbitrarily close to the original map in uniform convergence and in terms of the SobolevOrlicz norm. Further we show, in dimension 4 or higher, that such an approximation result cannot hold in Sobolev spaces W1,p where p is too small by constructing a sensepreserving 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 nonlinear elasticity. In this context a key property is their continuity. We show, by counterexample, 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 biLipschitz Sobolev map in Wk,p and composition of Lipschitz maps in Wk,p comparable with the classical inverse mapping theorem. As a...


Vlastnosti zobrazení s konečnou distorzí
Campbell, Daniel ; Hencl, Stanislav (advisor) ; Malý, Jan (referee)
We study the continuity of mappings of finite distortion, a set of mappings intended to model elastic deformations in nonlinear elasticity. We focus on continuity criteria for the innerdistortion function and prove that certain modulus of continuity estimates are sharp, i.e. cannot be im proved. We also give a proof of the continuity of mappings of finite distortion under simplified conditions on the integrability of the distortion function. 1


Optimalizace přistání na Měsíci
Campbell, Daniel ; Bárta, Tomáš (referee) ; Milota, Jaroslav (advisor)
Nazev prace: Optimali/ace pristani na Mesici Autor: Daniel Campbell Katedra. (listav): Katedra inatematicke analy/y Vedouci bakalafske prace: Doc.RNDr. .laroslav Miluta. CSc. email vedoueiho: Jaroslav.Milota'Q'mlT.cuni.c/ Abstrakt: V toto prat:i vytvoriine a /komnamo rnodol, kt.ory ])opisujo raketu pri pristani na inesiui. Urchin za jakych okolnosti l/,e pristat a zda f'xiistnji1 kontrol. klery by niininiili/oval inno/stvi jjotrcbneho paliva pon ziteho pri pristani. Pokud existuje. pak tento prvek najdome a doka/eme vlastnowl . Title:0ptimilisation of the moonlanding problem Author:Daniel Campbell Department:Katedra matematiuke analy/y Snpnrvisor:Doc. RNDr. Jaroslav Milota, CSc. Supervisor's email address:Jarosla\.Milota'(PniJr.(.:nni.c/ Abstract: In this paper \vo are to rreate and examine a model, which describes (he motion of a rocket landing on the surface of the moon. We will determine under which circumstances it is possible to make (.he landing and determine whether there exists some way of landing that minimises fuel consumption. IT so we are to find this method and prove the desired property.
