Soboljevljevi prostori i direktna metoda varijacijskog računa

master's thesis

• Title (english): Sobolev spaces and direct method of variational calculus
• Author: Borja Rukavina
• Mentor: Marko Erceg (mentor)
• Committee member: Marko Erceg (predsjednik povjerenstva)
• Committee member: Boris Muha (član povjerenstva)
• Committee member: Igor Pažanin (član povjerenstva)
• Committee member: Nenad Antonić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-09-30, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu se upoznajemo s osnovama teorije distribucija, Soboljevljevih prostora i varijacijskog računa u svrhu proučavanja slabih formulacija nekih klasičnih rubnih problema. Najprije motiviramo uvođenje distribucija te pokazujemo neka osnovna svojstva, a od posebne važnosti nam je pojam slabe derivacije. Pomoću njega definiramo Soboljevljeve prostore, za koje se pokazuje da su Banachovi. Zatim pokazujemo niz rezultata na Soboljevljevim prostorima od kojih se ističu Poincar´eova nejednakost, operatori proširenja i teorem o tragu. Potom proučavamo teoriju varijacijskog računa na apstraktnim refleksivnim Banachovim prostorima. Predstavljamo direktnu metodu varijacijskog računa i dajemo generalni rezultat. Dokazujemo poznatu Lax-Milgramovu lemu, koja se često koristi u dokazima egzistencije slabog rješenja. Za kraj navodimo nekoliko rubnih problema od interesa i primjenjujemo dane rezultate.

Abstract (english)

In this thesis, we present the fundamentals of theory of distributions, Sobolev spaces and calculus of variations with the objective of studying weak formulations of some classical boundary value problems. Firstly, we motivate the introduction of distributions and show some basic properties, where the notion of weak derivation will be of particular importance. We use it to define Sobolev spaces, which we show to be Banach spaces. We then present a series of results in Sobolev spaces, of which the Poincar´e inequality, the expansion operators, and the trace theorem bear the most significance. Next we study the theory of calculus of variations on abstract reflexive Banach spaces. We present the direct method of calculus of variations and give the general result. We prove the well-known Lax-Milgram lemma, which is often used in proofs of existence of a weak solution. Finally, we present some boundary value problems of interest and apply the mentioned results.

• Keywords: teorije distribucija; Soboljevljevi prostori; varijacijski račun; slaba derivacija; Banachovi prostori; Lax-Milgramovu lema
• Keywords (english): theory of distributions; Sobolev spaces; calculus of variations; Banach spaces; Lax-Milgram lemma
• Language: croatian
• URN:NBN: urn:nbn:hr:217:535823
• Study programme: Title: Applied Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2021-03-05 12:27:58