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Geometric Function Theory and its application in Nonlinear Elasticity
Bouchala, Ondřej ; Hencl, Stanislav (advisor) ; Pankka, Pekka (referee) ; Kružík, Martin (referee)
This thesis is divided into two parts. The first part focuses on mappings in Rn and the weak limits of homeomorphisms in the Sobolev space W1,p . Our primary concern is the concept of "injectivity almost everywhere". We demonstrate that when p ≤ n − 1, the weak limit of homeomorphisms can fail to satisfy this condition. Conversely, when p > n − 1, the weak limit is "injective almost everywhere". In the second part, we investigate the Hardy spaces in the complex plane. It is established that for a simply connected domain Ω ⊊ C, there exists a constant HΩ such that any conformal mapping from the unit disk in C onto Ω belongs to the Hardy space Hp for all p < HΩ. Conversely, for q > HΩ, no such mapping exists in the space Hq . However, we demonstrate that by allowing quasiconformal mappings instead of conformal ones, a quasiconformal mapping can be found from the unit disk onto Ω that belongs to the Hardy space Hp for every 0 < p < ∞. 1
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Measures of non-compactness of Sobolev embeddings
Bouchala, Ondřej ; Hencl, Stanislav (advisor)
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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Measures of non-compactness of Sobolev embeddings
Bouchala, Ondřej ; Hencl, Stanislav (advisor)
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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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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Construction of a von Koch snowflake
Bouchala, Ondřej ; Hencl, Stanislav (advisor) ; Vlasák, Václav (referee)
A Mapping from C onto C is quasiconformal, if it maps "infinitesimally small circles" onto "infinitesimally small ellipses". In other words, its real derivative in almost every point (which is for each point linear mapping from plane to plane) maps circles to ellipses with bounded ratio of axes. Koch snowflake is well-known inductively defined fractal: Using Beurling-Ahlfors extension we will prove, that there exists quasi- conformal mapping from the plane onto the plane, which maps unit disk onto Koch snowflake. 1
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