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.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-642-18363-8_13
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-18362-1
Online ISBN: 978-3-642-18363-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)
