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.
Aalto

Harnack Estimates for Supersolutions to a Nonlinear Degenerate Parabolic Equation

Tuomo Kuusi

Dissertation for the degree of Doctor of Philosophy to be presented with due permission of the Department of Engineering Physics and Mathematics for public examination and debate in Auditorium D at Helsinki University of Technology (Espoo, Finland) on the 16th of November, 2007, at 12 o'clock noon.

Dissertation in PDF format (ISBN 978-951-22-8975-2)   [438 KB]
Dissertation is also available in print (ISBN 978-951-22-8974-5)

Abstract

In this work, we prove both global and local weak Harnack estimates for supersolutions to a nonlinear degenerate parabolic partial differential equation using measure-theoretical arguments. The main tools are various estimates for both sub- and supersolutions, expansion of positivity, the comparison principle and the existence result for a Dirichlet problem with zero lateral boundary values and square-integrable initial data.

Keywords: parabolic nonlinear partial differential equations, Harnack estimates, weak solutions, parabolic Moser's method, De Giorgi's estimates, intrinsic time scale, regularity of solutions, Hölder continuity of solutions

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© 2007 Helsinki University of Technology

Last update 2011-05-26