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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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Nonlinear classes of mappings: properties and approximation
Doležalová, Anna ; Hencl, Stanislav (advisor) ; Pratelli, Aldo (referee) ; Krömer, Stefan (referee)
In this thesis, we explore classes of mappings suitable for models in Nonlinear Elastic- ity. We investigate whether, given the presence of certain desirable properties, there exists an element within the class that exhibits pathological behaviour. In the presented papers, we primarily focus on subclasses of Sobolev mappings, particularly weak closures of home- omorphisms with additional properties. These properties typically manifest themselves in the form of an additional term in the energy functional. We show that weak limits of Sobolev homeomorphisms in W1,n−1 satisfy the so-called (INV ) condition if the integrability of the reciprocals of the Jacobians is sufficiently high. This result is sharp and we present a counterexample for cases of lower integrability. The (INV ) condition is also preserved under weak limits when we add a term dependent on the cofactor matrix of the derivative, as its integrability provides some regularity for the inverses of the homeomorphisms in the sequence. Furthermore, we show that assumptions on regularity of the inverses can also ensure a.e. differentiability of the limit. Other topics investigated in this thesis include the sizes of critical sets violating the Luzin (N) condition in the case of Sobolev homeomorphisms and the (dis)continuity of mappings of generalized...
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Functional equations
Konopecký, František ; Hencl, Stanislav (advisor) ; Černý, Robert (referee)
Nazev pracc: Funkcionalni rovnice Autor: Frantisck Konopccky Katedra (ustav): Katcdra matcmaticke analyzy Vedouci bakalafske pracc: RNDr. Stanislav Hcncl, Ph.D. e-mail vedouciho: hencK(.i)karlni.mrT.cmri.cz Abstrakt: V pfcdlozcnc praci ye zabyvame mctodami fcscni funkcionalnich rovnic, triky, kterc fescni usnadnuji, vlastnostmi funkci, kterc pfi fescni pomahaji a myslenkovymi po- stupy pfi samotnem fescni. Nejprve za/Jvame pojem a praci s funkcionalni rovnici. Pote vysvetlnjeme dvc hlavni inetody - Substitucni-A Cauchyova. Naslcduji diilczite vlastnosti fnnkci a po nich ka]>itola o castych chybach pfi foscni. Ko konci se venujcme Cauchyove rovnici z vice stran a naslednje nckolik fesenych i nofcsenych pfikladu. Klicova slova: Fnnkcc, funkcionalni rovnice. Title: Functional Equations Author: Frantisek Konopccky Department; Department of Mathematical Analysis Supervisor: RNDr. Stanislav Hcncl, Ph.D. Supervisor's e-mail address: hencK(fikarlin.mfi.cnui.cz Abstract: In this thesis we study methods of solving strategies of functional equations, tricks, which make solving easier, special function properties, that help in solution and also ideas in solution itself. Firstly we get closer to term functional equation and show how to work with it. Then we explain two main methods Substitution method and Cauchy's...
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Analysis of equilibria in economic models
Stibůrek, David ; Pražák, Dalibor (advisor) ; Hencl, Stanislav (referee)
Nazev pracc: Analyza roviiovaznych bodu v ekonomickycli rnodelech Autor: David Stiburek Katcdra (ustav): Katedra matcmatickc analyzy Vcdouci bakalarske prace: UNDr. Dalibor Prazak, Ph.D. (- -mail vedouciho: ra'/akiojkarli Abstrakt: V pfcdlozcne praci studujcme rovnovahu v ekonomickycli mode- Icch. V fade ekonomickycli situacicli muzcrnc pozorovat, zc nastanc rovno- vazna situce neboli, ze od urciteho okamziku nenastauou zadne velke zmcny. V rnnoha disciplinach bychom si pfali znat tyto rovnovaznc hodnoty. Ovscm inohou byt i situce, kdy v uekterych oblaytech uemusi k rovnovaze vubec dojit. Proto v teto praci nejdfivc zkouinamc, kdy se situace vubec muze do- stat do neja,ke rovuovahy neboli, kdy jc stabilni. A pole sc pokousimo \ircit rovnovaznou hoduotu a take relaxacni cas. Klicova slova: Ilovnovalia, stabilita a relaxacni doba. Title: Aiialysia of equilibria in economic models Author: David Stiburek Department: Department of Mathemathical analysis Supervisor: RNDr. Dalibor Prazak, Ph.D. Supervisor's (vrnail address: pra/ak'iikarliii.nifF.cuiii.rz Abstract: In the present work we study equilibria in ecomomics models. In many states we can see something, what can be simple called ay steady state. In many branches we would like to know these equilibria. But there exist some situations, when we; can't expect...
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Differentiability of the inverse mapping
Konopecký, František ; Hencl, Stanislav (advisor) ; Honzík, Petr (referee)
Primary objective of the thesis is proof of the statement that if for ∈ ℕ a ≥ 1 a bilipschitz mapping belongs to +1, loc ∩ ,∞ loc then also its inverse −1 belongs to +1, loc . We prove a similar statement also for spaces loc . For this purpose we construct a new ordering of -th partial derivatives to generalized Jacobian matrix. Thanks to this matrix we are able to differentiate matrices in an applicable way. Generalized Jacobian matrix is projected so that there still holds the Chain rule and, in some way, also rules for matrices product differentiation. 1
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Nerovnosti pro integrální operátory
Holík, Miloslav ; Pick, Luboš (advisor) ; Hencl, Stanislav (referee)
The presented work contains a survey of the so far known results about the operator inequalities of the type "good λ", "better good λ" and "rearranged good λ" on the function spaces over the Euclidean space with the Lebesgue measure and their corollaries in the form of more complex operator inequal- ities and norm estimates. However, the main aim is to build similar theory for the Riesz potential operator on the function spaces over the quasi-metric space with the so-called "doubling" measure. Combining the corollaries of this theory with the known norm estimates we obtain the boundedness for the Riesz potential operator on the Lebesgue and Lorentz spaces.
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