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A new type of regularity with applications to the wave front sets

Nova vrsta regularnosti sa primenama na talasni front

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2016
IzvestajKomisije6883.pdf (399.2Kb)
Disertacija6883.pdf (660.8Kb)
Author
Tomić, Filip
Mentor
Teofanov, Nenad
Committee members
Pilipović, Stevan
Teofanov, Nenad
Prangoski, Bojan
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Abstract
We introduce a family of smooth functions which are "less regu-lar" than the Gevrey functions, and study its basic properties. In particular we prove the standard results concerning algebra property and stability under finite order derivation. Moreover, we  construct infnite order operators which leads us to the definition of class with ultradifferentiable property. We also prove that our classes are inverse-closed, and this result is the essential part in the proof of our main result presented in the final Chapter. Moreover, using the techniques of microlocal analysis, we introduce and investigate the corresponding wave front sets, and the prove the results related to singular support of a distribution. Our main results shows how the singularities of solutions to partial differential equations (PDE's in short) propagate in the framework of our regularity.
U ovoj tezi definišemo novu klasu glatkih funkcija i izučavamo njihove osnovne osobine. Pokazujemo da naše klase imaju svojsto algebre kao i da su zatvorene u odnosu na delovanje operatora izvoda konačnog reda.Sta više, konstruišemo diferencijalne operatore beskonačnog reda i to nas dovodi do definicije ultradiferencijabilnih klasa funkcija. Takode dokazujemo osobinu zatvorenosti u odnosu na inverze, i taj rezultat je najvažniji deo u dokazu glavne teoreme koja je formulisana u poslednjoj glavi. Koristeći tehnike mikrolokalne analize, uvodimo i izučavamo odgovarajuće talasne frontove, i pokazujemo odgovarajuća tvrdjenja vezana za singularni nosač distribucije. Naš glavni rezultat pokazuje kako se prostiru singulariteti rešenja linearnih parcijalnih diferencijalnih jednačina u okviru naše regularnosti.
Faculty:
Универзитет у Новом Саду, Природно-математички факултет
Date:
30-09-2016
Projects:
  • Methods of Functional and Harmonic Analysis and PDE with Singularities (RS-174024)
Keywords:
ultradifferentiable functions / ultradiferencijabilne funkcije / distribucije / talasni front / parcijalni diferencijalni operatori / singularni nosac / Klase Ževrea / distributions / wave front set / partila differential operators / singular support / Gevrey classes
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Handle
https://hdl.handle.net/21.15107/rcub_nardus_7187
URI
http://www.cris.uns.ac.rs/DownloadFileServlet/Disertacija146796803608997.pdf?controlNumber=(BISIS)101444&fileName=146796803608997.pdf&id=6334&source=NaRDuS&language=sr
https://nardus.mpn.gov.rs/handle/123456789/7187
http://www.cris.uns.ac.rs/record.jsf?recordId=101444&source=NaRDuS&language=sr
http://www.cris.uns.ac.rs/DownloadFileServlet/IzvestajKomisije146796805211222.pdf?controlNumber=(BISIS)101444&fileName=146796805211222.pdf&id=6335&source=NaRDuS&language=sr

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