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Author:Lehtelä, Pekka
Title:BV-funktiot metrisissä mitta-avaruuksissa
Functions of Bounded Variation on Metric Measure Spaces
Publication type:Master's thesis
Publication year:2013
Pages:iv + 52      Language:   fin
Department/School:Matematiikan ja systeemianalyysin laitos
Main subject:Matematiikka   (Mat-1)
Supervisor:Kinnunen, Juha
Instructor:
Electronic version URL: http://urn.fi/URN:NBN:fi:aalto-201309137661
OEVS:
Electronic archive copy is available via Aalto Thesis Database.
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Location:P1 Ark Aalto  1647   | Archive
Keywords:functions of bounded variation
variation measure
perimeter
isoperimetric inequality
coarea formula
structure theorem
BV-funktio
variaatiomitta
perimetri
isoperimetrinen epäyhtälö
koarekaava
struktuurilause
Abstract (eng): Functions of bounded variation, abbreviated as BV functions, define an important extension of the Sobolev functions, which is used in the calculus of variations and geometric measure theory.
In the Euclidean case, BV functions are locally integrable functions, whose weak first partial derivatives are Radon measures.
In the metric case BV functions are defined using relaxation as the closure of Lipschitz functions with respect to suitable convergence.
In a similar manner the variation measure of a function is defined.

An important part of the BV theory is the sets of finite perimeter, i.e. the sets whose characteristic functions are in BV.
Thus the variation measure of the characteristic function defines the perimeter measure of the set.
Main tools for the study of sets of finite perimeter are the isoperimetric inequality, which relates the measure of a set and its perimeter, and the coarea formula, which relates the variation measure of a function and the perimeters of its level sets.

The perimeter of a set gives, in a sense, a measure for the boundary of the set.
In the Euclidean case the perimeter of a set is given by n - 1 dimensional Hausdorff measure of the measure theoretic boundary of the set.
As a main result of the thesis, a corresponding structure theorem is given in the metric setting.
The theorem states that the perimeter of a set of finite perimeter is given by an integral of a bounded Borel function over the measure theoretic boundary of the set with respect to a Hausdorff type measure of codimension 1.
Abstract (fin): BV -funktiot (engl. bounded variation) muodostavat geometrisen mittateorian ja variaatiolaskennan kannalta tärkeän Sobolev-funktioavaruuden laajennuksen.
Euklidisessa avaruudessa BV -funktiot ovat lokaalisti integroituvia funktioita, joiden ensimmäiset heikot osittaisderivaatat ovat Radon-mittoja.
Metrisissä avaruuksissa BV -funktiot määritellään käyttäen relaksoitua määritelmää Lipschitz-funktioavaruuden sulkeumana sopivan suppenemisen suhteen.
Vastaavasti voidaan edelleen määritellä funktion variaatiomitta.

Tärkeän osan BV -funktioiden teoriaa muodostavat äärellisperimetriset joukot, eli joukot joiden karakteristiset funktiot ovat BV -funktioita ja siten joukon karakteristisen funktion variaatiomitta määrittää joukolle perimetrimitan. äärellisperimetristen joukkojen tutkimuksessa keskeisimpiä työkaluja ovat isoperimetrinen epäyhtälö, joka antaa yhteyden joukon mitan ja sen perimetrin välille, sekä koareakaava, joka antaa variaatiomitalle esityskaavan funktion tasojoukkojen perimetrien avulla.

Joukon perimetri kuvaa tietyssä mielessä joukon reunan mittaa.
Euklidisessa avaruudessa perimetrimitta saadaan joukon mittateoreettisen reunan n - 1 -ulotteisena Hausdorffin mittana.
Työn päätuloksena esitetään vastaava esityskaava metrisessä tapauksessa, jonka mukaan äärellisperimetrisen joukon perimetri saadaan rajoitetun Borel-funktion integraalina joukon mittateoreettisen reunan yli Hausdorff-tyyppisen mitan suhteen.
ED:2013-09-05
INSSI record number: 47166
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