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Sobolev embedding theorem on domains without Lipschitz boundary
Roskovec, Tomáš ; Hencl, Stanislav (advisor) ; Honzík, Petr (referee)
We study the Sobolev embeddings theorem and formulate modified theorems on domains with nonlipschitz boundary. The Sobolev embeddings the- orem on a domain with Lipschitz boundary claims f ∈ W1,p ⇒ f ∈ Lp∗ (p) , kde p∗ (p) = np n − p . The function p∗ (p) is continuous and even smooth. We construct a domain with nonlipschitz boundary and function of the optimal embedding i.e. analogy of p∗ (p) is not continous. In the first part, according to [1], we construct the domain with the point of discontinuity for p = n = 2. Though we used known construction of domain, we prove this by using more simple and elegant methods. In the second part of thesis we suggest the way how to generalize this model domain and shift the point of discontinuity to other point than p = n = 2.
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Properties of Sobolev Mappings
Roskovec, Tomáš ; Hencl, Stanislav (advisor)
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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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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Sobolev embedding theorem on domains without Lipschitz boundary
Roskovec, Tomáš ; Hencl, Stanislav (advisor) ; Honzík, Petr (referee)
We study the Sobolev embeddings theorem and formulate modified theorems on domains with nonlipschitz boundary. The Sobolev embeddings the- orem on a domain with Lipschitz boundary claims f ∈ W1,p ⇒ f ∈ Lp∗ (p) , kde p∗ (p) = np n − p . The function p∗ (p) is continuous and even smooth. We construct a domain with nonlipschitz boundary and function of the optimal embedding i.e. analogy of p∗ (p) is not continous. In the first part, according to [1], we construct the domain with the point of discontinuity for p = n = 2. Though we used known construction of domain, we prove this by using more simple and elegant methods. In the second part of thesis we suggest the way how to generalize this model domain and shift the point of discontinuity to other point than p = n = 2.
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