Regularity and convergence results in the calculus of variations on metric spaces
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Doctoral thesis (article-based)
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Author
Date
2007-03-02
Major/Subject
Mcode
Degree programme
Language
en
Pages
13, [88]
Series
Research reports / Helsinki University of Technology, Institute of Mathematics. A, 518
Abstract
This dissertation studies regularity, convergence and stability properties for minimizers of variational integrals on metric measure spaces. The treatise consists of four articles in which the Moser iteration, Harnack's inequality and Harnack's convergence principle are considered in connection with quasiminimizers of the p-Dirichlet integral. In addition, we study a nonlinear eigenvalue problem in this setting. This is done in metric spaces equipped with a doubling measure and supporting a weak (1,p)-Poincaré inequality.Väitöskirjassa tutkitaan säännöllisyys-, suppenemis- ja stabiilisuusominaisuuksia variaatio-ongelmien ratkaisuille metrisessä avaruudessa. Työ koostuu neljästä artikkelista, jotka käsittelevät muun muassa Moserin menetelmää, Harnackin epäyhtälöä ja Harnackin suppenemisperiaatetta p-Dirichlet-integraalin kvasiminimoijille. Lisäksi tarkastelemme niin sanottua epälineaarista ominaisarvo-ongelmaa. Työssä tutkitaan metristä avaruutta, jonka mitta on tuplaava ja jossa on voimassa heikko (1,p)-Poincarén epäyhtälö.Description
Keywords
Caccioppoli inequality, doubling measure, Harnack convergence theorem, Harnack inequality, Moser iteration, Newtonian space, nonlinear eigenvalue problem, p-Dirichlet integral, p-Laplace equation, Poincaré inequality, quasiminimizer, Rayleigh quotient, Sobolev space, subminimizer, superminimizer, Caccioppolin epäyhtälö, epälineaarinen ominaisarvo-ongelma, Harnackin suppenemisperiaate, Harnackin epäyhtälö, kvasiminimoija, Moserin iteraatio, Newtonin avaruus, p-Dirichlet-integraali, p-Laplacen yhtälö, Poincarén epäyhtälö, Rayleigh-osamäärä, Sobolevin avaruus, subminimoija, superminimoija, tuplaava mitta
Parts
- Marola, N., Moser's method for minimizers on metric measure spaces, Helsinki University of Technology, Institute of Mathematics, Research Report A478, 2004. [article1.pdf] © 2004 by author.
- Latvala, V., Marola, N. and Pere, M., Harnack's inequality for a nonlinear eigenvalue problem on metric spaces, Journal of Mathematical Analysis and Applications 321 (2006), 793-810.
- Björn, A. and Marola, N., Moser iteration for (quasi)minimizers on metric spaces, Manuscripta Mathematica 121 (2006), 339-366.
- Kinnunen, J., Marola, N. and Martio, O., Harnack's principle for quasiminimizers, Ricerche di Matematica, to appear.
