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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 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 W1,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 Wk,p and composition of Lipschitz maps in Wk,p comparable with the classical inverse mapping theorem. As a...
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Measures of non-compactness of Sobolev embeddings
Bouchala, Ondřej ; Hencl, Stanislav (advisor) ; Honzík, Petr (referee)
The measure of non-compactness is defined for any continuous mapping T : X Y between two Banach spaces X and Y as β(T) := inf { r > 0: T(BX) can be covered by finitely many open balls with radius r } . It can easily be shown that 0 ≤ β(T) ≤ ∥T∥ and that β(T) = 0, if and only if the mapping T is compact. My supervisor prof. Stanislav Hencl has proved in his paper that the measure of non-compactness of the known embedding W k,p 0 (Ω) → Lp∗ (Ω), where kp is smaller than the dimension, is equal to its norm. In this thesis we prove that the measure of non-compactness of the embedding between function spaces is under certain general assumptions equal to the norm of that embedding. We apply this theorem to the case of Lorentz spaces to obtain that the measure of non-compactness of the embedding Wk 0 Lp,q (Ω) → Lp∗,q (Ω) is for suitable p and q equal to its norm. 1
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Method of Lagrange multipliers in Calculus of Variations
Borák, Vojtěch ; Černý, Robert (advisor) ; Hencl, Stanislav (referee)
Tato práce řeší několik základních příkladů z variačního počtu a demonstruje prospěšnost zamyšlení se a případné pozměnění úlohy bez snížení dimenze za přítomnosti vazeb. Všechny úlohy jsou řešeny metodou Lagrangeových multipli- kátorů. Především v konečné dimenzi demonstruje hypotézu autora ohledně ne- snižování dimenze problému klasifikace definitnosti diferenciální formy druhých derivací a ukazuje jednak příklad, ve kterém je autorův nápad prospěšný, i pří- klad, kde svádí na scestí. 1
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Properties of derivative
Marková, Hana ; Zelený, Miroslav (advisor) ; Hencl, Stanislav (referee)
In the bachelor thesis we relate the concepts of derivative, the Darboux pro- perty and the function of the Baire class one. It is shown that each derivative has Darboux property and is of the Baire class one. Furthermore, we characterize the functions of the Baire class one using their associated sets. We introduce the concept of Zahorski classes and put them in connection with the functions of the Baire class one with the Darboux property. At the end of the thesis, we prove the Clarkson-Denjoy theorem.
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Properties of Sobolev Mappings
Roskovec, Tomáš ; Hencl, Stanislav (advisor) ; Björn, Anders (referee) ; Zürcher, Thomas (referee)
We study the properties of Sobolev functions and mappings, especially we study the violation of some properties. In the first part we study the Sobolev Embedding Theorem that guarantees W1,p (Ω) ⊂ Lp∗ (Ω) for some parameter p∗ (p, n, Ω). We show that for a general domain this relation does not have to be smooth as a function of p and not even continuous and we give the example of the domain in question. In the second part we study the Cesari's counterexample of the continuous mapping in W1,n ([−1, 1]n , Rn ) violating Lusin (N) condition. We show that this example can be constructed as a gradient mapping. In the third part we generalize the Cesari's counterexample and Ponomarev's counte- rexample for the higher derivative Sobolev spaces Wk,p (Ω, Rn ) and characterize the validity of the Lusin (N) condition in dependence on the parameters k and p and dimension. 1
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Typical continuous and integrable functions
Hruška, David ; Hencl, Stanislav (advisor) ; Pražák, Dalibor (referee)
In this thesis we use the Baire categories to define the concept of "typical functions". Then we prove several theorems generally asserting that a typical function from a space of functions having some nice property does not have a stronger property. In particular we prove that a typical continuous or Hölder continuous function is nowhere differentiable, a typical continuous monotone function does not satisfy the Luzin (N) condition and a typical integrable function is nowhere continuous. Powered by TCPDF (www.tcpdf.org)
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Sobolev mappings and Cantor type counterexamples
Fiala, Martin ; Hencl, Stanislav (advisor) ; Vybíral, Jan (referee)
Sobolev mappings and Cantor type counterexamples Author: Martin Fiala Supervisor: doc. RNDr. Stanislav Hencl, Ph.D. Abstract: The aim of this work is to show one of the general con- structions of the mappings, which can be used to create different coun- terexamples in the theory of Sobolev mappings. The construction is described in detail and then it is used for a number of examples. The last chapter is devoted to a slight generalization of this construction. 1
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Entropy numbers
Kossaczká, Marta ; Vybíral, Jan (advisor) ; Hencl, Stanislav (referee)
In this work we study entropy numbers of linear operators. We focus on entropy numbers of identities between real finite-dimensional sequence spaces and present detailed proofs of their estimates. Then we describe relation between entropy numbers of identities between real spaces and between complex spaces, which allows us to establish similar estimates for complex spaces. Powered by TCPDF (www.tcpdf.org)
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Three lakes problem
Šulc, Dominik ; Hencl, Stanislav (advisor) ; Vejnar, Benjamin (referee)
Cílem této práce je nalezení řešení problému tří jezer a podrobný d·kaz jeho správnosti. Problém tří jezer (Lakes of Wada) je úloha, která spočívá v sestrojení tří otevřených souvislých množin v rovině, které se neprotínají a mají společnou hranici. Ukážeme, že takové množiny existují a že kromě uvedených vlastností mohou být dokonce obloukově souvislé. 1
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