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Dirichlet Energy Integral and Laplace Equation

  • Lars Diening
  • Petteri Harjulehto
  • Peter Hästö
  • Michael Růžička
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2017)

Abstract

For a constant \( q \ \epsilon \ (1, \infty)\), the Dirichlet energy integral is \( \int\limits_{\Omega}|\nabla u (x)|^q dx \). The problem is to find a minimizer for the energy integral among all Sobolev functions with a given boundary value function. The Euler–Lagrange equation of this problem is the q-Laplace equation,\(div(\mid\bigtriangledown u \mid^{q-2}\bigtriangledown u )\, = \,0\), which has to be understand in the weak sense.

Keywords

Weak Solution Laplace Equation Unique Minimizer Variable Exponent Strong Maximum Principle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Lars Diening
    • 1
  • Petteri Harjulehto
    • 2
  • Peter Hästö
    • 3
  • Michael Růžička
    • 4
  1. 1.Institute of MathematicsLMU MunichMunichGermany
  2. 2.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland
  3. 3.Department of Mathematical SciencesUniversity of OuluOuluFinland
  4. 4.Institute of MathematicsUniversity of FreiburgFreiburgGermany

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