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Dirichlet Energy Integral and Laplace Equation

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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Correspondence to Lars Diening .

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© 2011 Springer-Verlag Berlin Heidelberg

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Diening, L., Harjulehto, P., Hästö, P., Růžička, M. (2011). Dirichlet Energy Integral and Laplace Equation. In: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics(), vol 2017. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18363-8_13

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