Primjena nelinearne funkcionalne analize u teoriji elastičnosti

master's thesis

• Title (english): Application of nonlinear functional analysis in elasticity theory
• Author: Dominik Šime Samac
• Mentor: Josip Tambača (mentor)
• Committee member: Josip Tambača (predsjednik povjerenstva)
• Committee member: Matko Ljulj (član povjerenstva)
• Committee member: Igor Pažanin (član povjerenstva)
• Committee member: Slaven Kožić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-03-02, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovome radu iskazan je i dokazan John Ballov teorem o egzistenciji minimizatora određenih vrsta funkcionala. Kako bi se dokazao teorem, razvijena je teorija koja pokriva funkcionalnu analizu, s naglaskom na Banach-Saks-Mazurov teorem i Banach-Eberlein-Šmulianov teorem. Uvedeni su Soboljevljevi prostori na kojim se traže minimizatori te su dokazana osnovna svojstva Soboljevljevih prostora neophodna za dokazivanje kasnijih teorema. Prije završnih rezultata, napravljen je pregled nekih tema iz nelinearne funkcionalne analize gdje se definirala Frechetova derivacija. Naglasak tog poglavlja je na Piolinom identitetu koji se nekoliko puta iskoristio u kasnijim teoremima. Posljednje poglavlje kreće s egzistencijom minimizatora na općenitim funkcionalima. S razvijanjem teorije određene pretpostavke na funkcionale su nadodane kako bi promatrani matematički model iz teorije elastičnosti imao smisla. Naposljetku, John Ballov teorem riješi pitanje pronalaska minimizatora i za takve funkcionale.

Abstract (english)

In this paper, John Ball's theorem on the existence of minimizers of certain types of functionals is stated and proved. In order to prove the theorem, a theory covering functional analysis is developed, with an emphasis on the Banach-Saks-Mazur theorem and the Banach-Eberlein-Šmulian theorem. Sobolev spaces are introduced, on which minimizers are sought, and the basic properties of Sobolev spaces necessary for proving later theorems are proved. Before the final results, an overview of some topics from nonlinear functional analysis where the Frechet derivative was defined was made. The emphasis of that chapter is on Piola's identity, which was used several times in later theorems. The last chapter starts with the existence of minimizers on general functionals. With the development of the theory, certain assumptions were added to the functional ones in order for the observed mathematical model from the theory of elasticity to make sense. Finally, John Ball's theorem solves the problem of finding a minimizer for such functionals as well.

• Keywords: John Ballov teorem o egzistenciji; Banach-Saks-Mazurov teorem; Banach-Eberlein-Šmulianov teorem; Soboljevljevi prostori; Frechetova derivacija
• Keywords (english): John Ball's existence theorem; Banach-Saks-Mazur theorem; Banach-Eberlein-Shmulian theorem; Soboljev premises; Frechet's derivation
• Language: croatian
• URN:NBN: urn:nbn:hr:217:804260
• Study programme: Title: Pure 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: 2023-08-18 09:46:39