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.
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Global Higher Integrability for Nonlinear Parabolic Partial Differential Equations in Nonsmooth Domains
Mikko Parviainen
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 November, 2007, at 12 noon.
Dissertation in PDF format (ISBN 978-951-22-8940-0) [372 KB]
Dissertation is also available in print (ISBN 978-951-22-8939-4)
Abstract
This thesis studies the global regularity theory for degenerate nonlinear parabolic partial differential equations. Our objective is to show that weak solutions belong to a higher Sobolev space than assumed a priori if the complement of the domain satisfies a capacity density condition and if the boundary values are sufficiently smooth. Moreover, we derive integrability estimates near the lateral and initial boundaries. The results of the thesis extend to parabolic systems as well. The higher integrability estimates provide a useful tool in several applications.
Keywords: boundary value problem, Caccioppoli inequality, capacity density, Gehring lemma, Giaquinta-Modica lemma, initial value problem, integrability of the gradient, nonlinear parabolic system, parabolic p-Laplace equation, reverse Hölder inequality
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© 2007 Helsinki University of Technology
Last update 2011-05-26