The doctoral dissertations of the former Helsinki University of Technology (TKK) and Aalto University Schools of Technology (CHEM, ELEC, ENG, SCI) published in electronic format are available in the electronic publications archive of Aalto University - Aaltodoc.

Regularity and Convergence Results in the Calculus of Variations on Metric Spaces

Niko Marola

Dissertation for the degree of Doctor of Science in Technology to be presented with due permission of the Department of Engineering Physics and Mathematics for public examination and debate in Auditorium E at Helsinki University of Technology (Espoo, Finland) on the 2nd of March, 2007, at 12 noon.

Overview in PDF format (ISBN 978-951-22-8597-6)   [204 KB]
Dissertation is also available in print (ISBN 978-951-22-8596-9)


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.

This thesis consists of an overview and of the following 4 publications:

  1. Marola, N., Moser's method for minimizers on metric measure spaces, Helsinki University of Technology, Institute of Mathematics, Research Report A478, 2004. © 2004 by author.
  2. 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.
  3. Björn, A. and Marola, N., Moser iteration for (quasi)minimizers on metric spaces, Manuscripta Mathematica 121 (2006), 339-366.
  4. Kinnunen, J., Marola, N. and Martio, O., Harnack's principle for quasiminimizers, Ricerche di Matematica, to appear.

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

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© 2007 Helsinki University of Technology

Last update 2011-05-26