Dirichlet Energy Integral and Laplace Equation

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.