search query: @keyword equation / total: 2
reference: 1 / 2
« previous | next »
Author:Vuojamo, Vesa
Title:Harnackin epäyhtälö
Harnack's Inequality
Publication type:Master's thesis
Publication year:2012
Pages:[4] + 42      Language:   fin
Department/School:Matematiikan ja systeemianalyysin laitos
Main subject:Matematiikka   (Mat-1)
Supervisor:Kinnunen, Juha
Instructor:
OEVS:
Electronic archive copy is available via Aalto Thesis Database.
Instructions

Reading digital theses in the closed network of the Aalto University Harald Herlin Learning Centre

In the closed network of Learning Centre you can read digital and digitized theses not available in the open network.

The Learning Centre contact details and opening hours: https://learningcentre.aalto.fi/en/harald-herlin-learning-centre/

You can read theses on the Learning Centre customer computers, which are available on all floors.

Logging on to the customer computers

  • Aalto University staff members log on to the customer computer using the Aalto username and password.
  • Other customers log on using a shared username and password.

Opening a thesis

  • On the desktop of the customer computers, you will find an icon titled:

    Aalto Thesis Database

  • Click on the icon to search for and open the thesis you are looking for from Aaltodoc database. You can find the thesis file by clicking the link on the OEV or OEVS field.

Reading the thesis

  • You can either print the thesis or read it on the customer computer screen.
  • You cannot save the thesis file on a flash drive or email it.
  • You cannot copy text or images from the file.
  • You cannot edit the file.

Printing the thesis

  • You can print the thesis for your personal study or research use.
  • Aalto University students and staff members may print black-and-white prints on the PrintingPoint devices when using the computer with personal Aalto username and password. Color printing is possible using the printer u90203-psc3, which is located near the customer service. Color printing is subject to a charge to Aalto University students and staff members.
  • Other customers can use the printer u90203-psc3. All printing is subject to a charge to non-University members.
Location:P1 Ark Aalto  225   | Archive
Keywords:parabolic
nonlinear
partial
differential
equation
regularity
Moser-iteration
parabolinen
epälineaarinen
osittaisdifferentiaaliyhtälö
säännöllisyys
Moser-iteraatio
Abstract (eng): In the theory of partial differential equations there are only few simple equations that can be given an explicit formula for solutions.
In general equations are studied by proving certain existence and regularity results.
Smoothness and boundedness are examples of regularity results for solutions.
Harnack's inequality is a result which states that the essential supremum and infimum of a solution are comparable.
This means that an upper bound for the essential supremum can be obtained from the infimum by multiplying it by a constant.
This constant may only depend on the geometry of the sets considered and on the structure of the equation.

The proof of Harnack's inequality is based on Caccioppoli-type estimates and Moser's iteration.
This iteration leads to separate estimates for both supremum and infimum of solutions.
These estimates can be combined to give the Harnack's inequality by means of using parabolic John-Nirenberg lemma or a measure theoretical Bombieri's lemma.
Bombieri's lemma is considerably less technical than the parabolic version of John-Nirenberg lemma.

In this thesis a proof is given for Harnack's inequality for a certain type of parabolic partial differential equations.
A similar result was previously proven in an article by N.
Trudinger in 1968.
Trudinger's proof is simplified by replacing the parabolic John-Nirenberg lemma with Bombieri's lemma.
Calculations are also carried out in more detail.
Abstract (fin): Osittaisdifferentiaaliyhtälöiden teoriassa vain harvoille sopivan yksinkertaisille yhtälöille voidaan johtaa varsinainen ratkaisukaava.
Yleisesti yhtälöitä voidaan tarkastella erilaisten olemassaolo- sekä säännöllisyystulosten avulla.
Säännöllisyystuloksia ovat esimerkiksi ratkaisuiden sileys sekä rajoittuneisuus.
Harnackin epäyhtälö on tulos, joka kertoo, että yhtälön ratkaisuiden oleelliset supremum sekä infimum ovat verrannollisia keskenään vakiolla, joka riippuu vain tarkasteltavan joukon geometriasta sekä yhtälön rakenteesta.

Harnackin epäyhtälön johto perustuu Caccioppoli-tyyppisiin estimaatteihin sekä Moserin iteraatioon.
Tämän iteraation avulla ratkaisuille saadaan erikseen arviot sekä supremumille että infimumille.
Näiden arvioiden yhdistämiseen Harnackin epäyhtälöksi voidaan käyttää joko parabolista John-Nirenbergin lemmaa tai mittateoreettista Bombierin abstraktia lemmaa.
Koska Bombierin lemma on vain mittateoreettinen tulos, se ei ole yhtä tekninen kuin parabolisen BMO:n avulla johdettu John-Nirenbergin lemma.

Tässä työssä on johdettu Harnackin epäyhtälö tietyntyyppisille parabolisille osittaisdifferentiaaliyhtälöille.
Samanlainen tulos esiintyi alun perin N.
Trudingerin artikkelissa vuonna 1968.
Trudingerin käyttämä parabolinen John-Nirenbergin lemma on kuitenkin korvattu Bombierin lemmalla, mikä yksinkertaistaa todistusta.
Käytettyjen argumenttien yksityiskohdat on myös laskettu selkeämmin auki.
ED:2012-07-10
INSSI record number: 45017
+ add basket
« previous | next »
INSSI