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Given the $p$-Laplace equation, with $p>2$, which writes

\begin{align} -\text{div}(|\nabla{u}|^{p-2}\nabla u) = f \end{align}

I want to associate the functional $J(u)$ to be minimized in order to solve it.

On the notes I've written that the natural functional to associate is the $p$-Dirichlet energy which is:

\begin{align} J(u) = \int_{\Omega} \left( \frac{|\nabla u|^p}{p} - fu \right) dx \end{align}

The problem is that I can't see how such a functional is related with the equation. I've seen that

$$\nabla \left( \frac{|\nabla u|^p}{p} \right) = |\nabla{u}|^{p-2}\nabla u$$

but I don't understand how to pass from the equation to the functional. I intuitively understand that we want that solving $\nabla J(u) = 0$ is like solving the equation, but I cant' work out the steps to prove this relation. Maybe I should use Green's Formula or other vector identities, but how?

Matthew Cassell
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Dadeslam
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  • First, it can be really difficult to construct a functional, so don't worry if it doesn't come straight away. Secondly, you don't want to compute $\nabla J$, your PDE should be a 'critical point' of the variation of $J$, $\delta J/\delta u$. See here for more details. – Matthew Cassell Jun 26 '19 at 05:30
  • Is the thing you pointed out the Fréchet derivative with respect to the direction given by a test function $v$? – Dadeslam Jun 26 '19 at 06:57
  • My problem now is how to compute the functional derivative of $J$ along the direction of an arbitrary $v\in H^1_0$. I have just posted a new question about it https://math.stackexchange.com/questions/3274730/write-the-functional-derivative-of-the-dirichlet-energy – Dadeslam Jun 26 '19 at 09:12

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