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

Nonlinear Potential Theory of Elliptic Equations with Nonstandard Growth

Teemu Lukkari

Dissertation for the degree of Doctor of Science in Technology to be presented with due permission of the Faculty of Information and Natural Sciences for public examination and debate in Auditorium G at Helsinki University of Technology (Espoo, Finland) on the 7th of March, 2008, at 12 noon.

Overview in PDF format (ISBN 978-951-22-9240-0)   [220 KB]
Dissertation is also available in print (ISBN 978-951-22-9239-4)

Abstract

We consider the nonlinear potential theory of elliptic partial differential equations with nonstandard structural conditions. In such a theory, Harnack inequalities and the class of superharmonic functions related to the equation under consideration have a crucial role. We develop a technique for proving Harnack type inequalities to handle possibly unbounded solutions. After this, we show that the basic properties of the related superharmonic functions are similar to the case of standard structural conditions, and give applications of Harnack inequalities and superharmonicity. These include removability, growth of fundamental solutions, and superharmonic functions as solutions of equations involving measures.

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

  1. P. Harjulehto, J. Kinnunen, and T. Lukkari. Unbounded supersolutions of nonlinear equations with nonstandard growth. Boundary Value Problems 2007: Article ID 48348, 20 pages, 2007. © 2008 by authors.
  2. P. Harjulehto, P. Hästö, M. Koskenoja, T. Lukkari, and N. Marola. An obstacle problem and superharmonic functions with nonstandard growth. Nonlinear Analysis: Theory, Methods & Applications 67 (12): 3424-3440, 2007.
  3. T. Lukkari. Singular solutions of elliptic equations with nonstandard growth. Mathematische Nachrichten, to appear.
  4. T. Lukkari. Elliptic equations with nonstandard growth involving measures. Hiroshima Mathematical Journal, to appear. © 2008 Hiroshima University, Department of Mathematics. By permission.

Keywords: nonstandard growth, variable exponent, p(x)-Laplacian, logarithmic Hölder continuity, Caccioppoli estimate, Moser iteration, Harnack's inequality, regularity, comparison principle, superharmonic function, removability, growth of solutions, existence of generalized solutions, measure data

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© 2008 Helsinki University of Technology

Last update 2011-05-26