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Isoperimetric problem, Sobolev spaces and the Heisenberg group
Franců, Martin ; Pick, Luboš (advisor) ; Cianchi, Andrea (referee) ; Nekvinda, Aleš (referee)
In this thesis we study embeddings of spaces of functions defined on Carnot- Carathéodory spaces. Main results of this work consist of conditions for Sobolev- type embeddings of higher order between rearrangement-invariant spaces. In a special case when the underlying measure space is the so-called X-PS domain in the Heisenberg group we obtain full characterization of a Sobolev embedding. The next set of main results concerns compactness of the above-mentioned em- beddings. In these cases we obtain sufficient conditions. We apply the general results to important particular examples of function spaces. In the final part of the thesis we present a new algorithm for approximation of the least concave majorant of a function defined on an interval complemented with the estimate of the error of such approximation. 1
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Isoperimetric problem, Sobolev spaces and the Heisenberg group
Franců, Martin ; Pick, Luboš (advisor) ; Cianchi, Andrea (referee) ; Nekvinda, Aleš (referee)
In this thesis we study embeddings of spaces of functions defined on Carnot- Carathéodory spaces. Main results of this work consist of conditions for Sobolev- type embeddings of higher order between rearrangement-invariant spaces. In a special case when the underlying measure space is the so-called X-PS domain in the Heisenberg group we obtain full characterization of a Sobolev embedding. The next set of main results concerns compactness of the above-mentioned em- beddings. In these cases we obtain sufficient conditions. We apply the general results to important particular examples of function spaces. In the final part of the thesis we present a new algorithm for approximation of the least concave majorant of a function defined on an interval complemented with the estimate of the error of such approximation. 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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