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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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