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master's thesis
Teoremi ulaganja Soboljevljevih prostora i primjene
Zagreb: University of Zagreb, Faculty of Science, 2014. urn:nbn:hr:217:771305
Radošević, Ana
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Radošević, A. (2014). Teoremi ulaganja Soboljevljevih prostora i primjene (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:771305

Radošević, Ana. "Teoremi ulaganja Soboljevljevih prostora i primjene." Master's thesis, University of Zagreb, Faculty of Science, 2014. https://urn.nsk.hr/urn:nbn:hr:217:771305

Radošević, Ana. "Teoremi ulaganja Soboljevljevih prostora i primjene." Master's thesis, University of Zagreb, Faculty of Science, 2014. https://urn.nsk.hr/urn:nbn:hr:217:771305

Radošević, A. (2014). 'Teoremi ulaganja Soboljevljevih prostora i primjene', Master's thesis, University of Zagreb, Faculty of Science, accessed 22 March 2025, https://urn.nsk.hr/urn:nbn:hr:217:771305

Radošević A. Teoremi ulaganja Soboljevljevih prostora i primjene [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2014 [cited 2025 March 22] Available at: https://urn.nsk.hr/urn:nbn:hr:217:771305

A. Radošević, "Teoremi ulaganja Soboljevljevih prostora i primjene", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2014. Available at: https://urn.nsk.hr/urn:nbn:hr:217:771305

Metadata
TitleTeoremi ulaganja Soboljevljevih prostora i primjene
AuthorAna Radošević
MentorMladen Jurak (mentor)
Committee memberMladen Jurak (predsjednik povjerenstva)
Committee memberMaja Starčević (član povjerenstva)
Committee memberMarko Vrdoljak (član povjerenstva)
Committee memberNenad Antonić (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2014-12-08, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
U ovom radu dokazali smo teoreme ulaganja Soboljevljevih prostora Wk,q u Lp prostore (ili čak Hölderove prostore, ako je kq>n, gdje je n dimenzija prostora Rn) na ograničenim (dovoljno glatkim) domenama. Među njima je posebno važan Rellich-Kondrachovljev teorem, koji osigurava kompaktnost određenih ulaganja, što smo u dva primjera iskoristili za dokaz egzistencije rješenja nelinearnih rubnih zadaća. Prvi je primjer kvazilinearna eliptička PDJ s homogenim... More rubim uvjetom {−Δu+b(Du)+μu=0na Ωu=0na ∂Ω, gdje je Ω ograničena domena klase C2. Pretpostavili smo da je b:Rn→R Lipschitzova te zadovoljava uvjet na rast |b(p)|≤C(|p|+1) za neki C i za sve p∈Rn. Egzistencija rješenja u∈H2(Ω)∩H10(Ω) ove zadaće pokazana je pomoću Schauderovog teorema o fiksnoj točki, čija je pretpostavka kompaktnosti odgovarajućeg operatora osigurana kompaktnošću ulaganja H2(Ω)⊂H1(Ω). Drugi je primjer stacionarna Navier-Stokesova zadaća, također s homogenim rubnim uvjetom, na ograničenoj glatkoj domeni Ω, {−μΔu+(Du)u+grad p=fna Ωdivu=0na Ωu=0na Γ, gdje su vektorska funkcija u=(u1,…,un) i skalarna funkcija p tražene funkcije, dok je f zadana. Ovdje je egzistencija rješenja pokazana pomoću Galerkinove metode, dakle konstruirali smo ograničen niz aproksimativnih rješenja te na limesu nekog njegovog podniza dobili rješenje. Za to nam je bila potrebna kompaktnost ulaganja H10(Ω)⊂L2(Ω). Less
Abstract (english)
In this thesis we proved the Sobolev embedding theorems of spaces Wk,q(Ω) in Lp spaces (or even Hölder spaces, when kq>n, where n is the dimension of Rn) on bounded (smooth enough) domains. Among them, the Rellich-Kondrachov Theorem is particularly important, since it provides compactness of certain embeddings, which we have used in two examples to prove the existence of solutions of nonlinear boundary problems. The first example is the... More quasilinear elliptic PDE with homogeneous boundary condition {−Δu+b(Du)+μu=0on Ωu=0on ∂Ω, where Ω is a bounded domain of class C2. We assumed that b:Rn→R is Lipschitz continous and satisfies the growth condition |b(p)|≤C(|p|+1) for some C and all p∈Rn. The existence of a solution u∈H2(Ω)∩H10(Ω) of this problem is shown using Schauder’s Fixed Point Theorem, whose assumption of compactness of the corresponding operator is secured by the compactness of the embedding H2(Ω)⊂H1(Ω). The second example is the stationary Navier-Stokes problem, also with homogeneous boundary condition, on a bounded smooth domain Ω, {−μΔu+(Du)u+grad p=fon Ωdivu=0on Ωu=0on Γ, where the vector function u=(u1,…,un) and the scalar function p are required functions, while f is given. Here the existence of a solution is shown using the Galerkin method, i.e. we have constructed a bounded sequence of approximate solutions and got the solution as the limit of its subsequence. To do this, we needed the compactness of the embedding H10(Ω)⊂L2(Ω). Less
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:771305
Study programmeTitle: Applied Mathematics
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2019-02-01 11:54:04
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