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Author:Lahti, Panu
Title:Rajoitetusti heilahtelevien funktioiden Lebesguen lause
Lebesgue theorem for functions of bounded variation
Publication type:Master's thesis
Publication year:2011
Pages:61      Language:   fin
Department/School:Matematiikan ja systeemianalyysin laitos
Main subject:Matematiikka   (Mat-1)
Supervisor:Kinnunen, Juha
Instructor:Kinnunen, Juha
OEVS:
Electronic archive copy is available via Aalto Thesis Database.
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Location:P1 Ark Aalto  101   | Archive
Keywords:bounded variation
variation measure
perimeter measure
reduced boundary
measure theoretic boundary
structure theorem
Lebesgue theorem
rajoitettu heilahtelu
variaatiomitta
perimetrimitta
redusoitu reuna
mittateoreettinen reuna
struktuurilause
Lebesguen lause
Abstract (eng): Functions of bounded variation, abbreviated BV functions, are locally integrable functions whose weak first partial derivatives are Radon measures.
Thus they form a more general class of functions than Sobolev functions, whose weak first partial derivatives are locally integrable functions.
Some of the most central results derived for BV functions include a compactness result, the coarea formula, and versions of the Sobolev and Poincaré inequalities.

The characteristic functions of so-called sets of finite perimeter form an interesting special case of BV functions.
For these sets we can define the reduced boundary, which is a subset of the topological boundary.
It turns out that in the neighborhood of the reduced boundary the set resembles a half space in a measure theoretic sense.

Further, we can define the measure theoretic boundary of a set.
This is also a subset of the topological boundary and closely resembles the reduced boundary in a measure theoretic sense.
Utilizing this and other minor results we can prove a strong result about the structure of the reduced boundary: it is made up of compact subsets of smooth hyper-surfaces.
In addition, the Radon measure that acts as the derivative of the set of finite perimeter is simply the Hausdorff measure restricted to the reduced boundary.

According to the co-area formula, the level sets of BV functions are sets of finite perimeter.
This enables us to apply the aforementioned results to general BV functions.
It turns out that BV functions are (with the choice of a suitable representative) measure theoretically continuous apart from "jumps" over smooth hyper-surfaces.
This is expressed in an exact manner in the Lebesgue theorem for BV functions.
The result is substantially stronger than the Lebesgue theorem for functions that are merely integrable, but weaker than the corresponding result for Sobolev functions.
Abstract (fin): Rajoitetusti heilahtelevat funktiot eli BV-funktiot (engl. bounded variation) ovat lokaalisti integroituvia funktioita, joiden ensimmäisen kertaluvun heikot osittaisderivaatat ovat Radon-mittoja.
Ne muodostavat siis yleisemmän funktioluokan kuin Sobolevin funktiot, joiden ensimmäisen kertaluvun heikot osittaisderivaatat ovat lokaalisti integroituvia funktioita.
Keskeisimpiä BV-funktioille päteviä tuloksia ovat kompaktisuustulos, coarea-kaava sekä Sobolevin ja Poincarén epäyhtälöiden versiot.

Mielenkiintoisen BV-funktioiden erikoistapauksen muodostavat niin sanottujen äärellisperimetristen joukkojen karakteristiset funktiot.
Tällaisille joukoille voidaan määritellä redusoitu reuna, joka on topologisen reunan osajoukko.
Osoittautuu, että lähellä redusoitua reunaa joukko muistuttaa mittateoreettisessa mielessä puoliavaruutta.

Edelleen voidaan määritellä joukon mittateoreettinen reuna, joka on myös topologisen reunan osajoukko ja muistuttaa mittateoreettisessa mielessä hyvin paljon redusoitua reunaa.
Muun muassa tätä tietoa hyödyntäen voidaan todistaa vahva tulos redusoidun reunan rakenteesta: se koostuu sileiden hyperpintojen kompakteista osajoukoista.
Lisäksi äärellisperimetrisen joukon derivaattana toimiva Radon-mitta on itse asiassa vain Hausdorffin mitta rajoitettuna redusoidulle reunalle.

Coarea-kaavan mukaan BV-funktion tasojoukot ovat äärellisperimetrisiä joukkoja, mikä mahdollistaa mainittujen tulosten soveltamisen yleisiin BV-funktioihin.
Osoittautuu, että BV-funktiot ovat (sopiva edustaja valiten) mittateoreettisesti jatkuvia lukuun ottamatta "hyppyjä" yli sileiden hyperpintojen.
Täsmällisesti tämä tulee ilmaistuksi BV-funktioiden Lebesguen lauseessa.
Tulos on olennaisesti vahvempi kuin pelkästään integroituville funktioille saatava Lebesguen lause, joskin heikompi kuin Sobolevin funktioille saatava.
ED:2011-09-22
INSSI record number: 42796
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