In this post, we would like to discuss some combinatorial aspects of the $p$-Laplacian. Namely, let $\int_\Omega |\nabla u|^p \, dx$ be the $p$-Dirichlet energy, where $u \in W^{1,p}(\Omega)$ and $p>1$. Its first variation is given by

where $\xi_1 \in W^{1,p}(\Omega)$.

The second variation (if exists) is also easy to compute:

\begin{align} \notag D^2 \left(\int_\Omega |\nabla u|^p \, dx\right) (\xi_1, \xi_2) &= p(p-2) \int_\Omega |\nabla u|^{p-4} (\nabla u, \nabla \xi_1) (\nabla u, \nabla \xi_2) \, dx \newline \notag &+ p \int_\Omega |\nabla u|^{p-2} (\nabla \xi_1, \nabla \xi_2) \, dx, \end{align}

where $\xi_2 \in W^{1,p}(\Omega)$.

Let us make some effort to calculate the third variation:

\begin{align} \notag D^3 \left(\int_\Omega |\nabla u|^p \, dx\right) (\xi_1, \xi_2, \xi_3) &= p(p-2)(p-4) \int_\Omega |\nabla u|^{p-6} (\nabla u, \nabla \xi_1) (\nabla u, \nabla \xi_2) (\nabla u, \nabla \xi_3) \, dx \newline \notag &+ p(p-2) \int_\Omega |\nabla u|^{p-4} (\nabla u, \nabla \xi_1) (\nabla \xi_2, \nabla \xi_3) \, dx \newline \notag &+ p(p-2) \int_\Omega |\nabla u|^{p-4} (\nabla u, \nabla \xi_2) (\nabla \xi_1, \nabla \xi_3) \, dx \newline \notag &+ p(p-2) \int_\Omega |\nabla u|^{p-4} (\nabla u, \nabla \xi_3) (\nabla \xi_1, \nabla \xi_2) \, dx, \end{align}

where $\xi_3 \in W^{1,p}(\Omega)$.

We already start seeing some structure. So, let us now try to derive a general formula for the $n$-th variation of the energy functional. Our main result is the following one.

Proposition. Let $u \in W^{1,p}(\Omega)$. If for a natural $n \geq 1$ there exists $n$-th variation of the $p$-Direchlet energy of $u$ in direction $(\xi_1,\dots,\xi_n) \in (W^{1,p}(\Omega))^n$, then

\begin{align} \notag &D^n \left(\int_\Omega |\nabla u|^p \, dx\right) (\xi_1, \dots, \xi_n) \newline \notag &= \int_\Omega \left( \sum\limits_{i=0}^{\lfloor \frac{n}{2} \rfloor} |\nabla u|^{p-2(n-i)} \prod\limits_{j=0}^{n-i-1} (p-2j) \left[\sum\limits_{\sigma \in B(n,n-2i)} \prod\limits_{k=1}^{n-2i} (\nabla u, \nabla \xi_{\sigma(k)}) \left(\sum\limits_{\omega \in P(n,\sigma)} \prod\limits_{l=1}^{i} (\nabla \xi_{\omega(l,1)}, \nabla \xi_{\omega(l,2)}) \right) \right] \right) dx, \end{align}

where

• $B(n,n-2i)$ is the set of all possible $(n-2i)$-combinations of $\{1,2,\dots,n\}$ such that the ordering inside each $\sigma \in B(n,n-2i)$ is immaterial. Evidently, the cardinality of $B(n,n-2i)$ is ${n \choose n-2i}$. In particular, if $i=0$, then $card(B(n,n-2i)) = 1$.

• $P(n,\sigma)$ is the set of all possible partitions of the set $\{1,2,\dots,n\} \setminus \sigma$ into pairs such that the ordering of pairs and inside a pair is immaterial. Note that $card(\sigma)=n-2i$, and hence the number of pairs in each $\omega \in P(\sigma)$ is $i$. It is not hard to see that the cardinality of $P(\sigma)$ is $\frac{(2i)!}{2^i i!}$. We represent $\omega$ as a $i \times 2$-matrix $(\omega(s,t))_{s=1..i,~t=1,2}$. For instance, if $n=6$ and $\sigma = \{1,2\}$, then

Let us also calculate the fourth variation (just for fun) - either by straightforward computation, or by application of our general formula:

\begin{align} \notag D^4 &\left(\int_\Omega |\nabla u|^p \, dx\right) (\xi_1, \xi_2, \xi_3) \\ \notag &= p(p-2)(p-4)(p-6) \int_\Omega |\nabla u|^{p-8} (\nabla u, \nabla \xi_1) (\nabla u, \nabla \xi_2) (\nabla u, \nabla \xi_3) (\nabla u, \nabla \xi_4) \, dx \\ \notag &+ p(p-2)(p-4) \int_\Omega |\nabla u|^{p-6} (\nabla u, \nabla \xi_1) (\nabla u, \nabla \xi_2) (\nabla \xi_3, \nabla \xi_4) \, dx \\ \notag &+ p(p-2)(p-4) \int_\Omega |\nabla u|^{p-6} (\nabla u, \nabla \xi_1) (\nabla u, \nabla \xi_3) (\nabla \xi_2, \nabla \xi_4) \, dx \\ \notag &+ p(p-2)(p-4) \int_\Omega |\nabla u|^{p-6} (\nabla u, \nabla \xi_1) (\nabla u, \nabla \xi_4) (\nabla \xi_2, \nabla \xi_3) \, dx \\ \notag &+ p(p-2)(p-4) \int_\Omega |\nabla u|^{p-6} (\nabla u, \nabla \xi_2) (\nabla u, \nabla \xi_3) (\nabla \xi_1, \nabla \xi_4) \, dx \\ \notag &+ p(p-2)(p-4) \int_\Omega |\nabla u|^{p-6} (\nabla u, \nabla \xi_2) (\nabla u, \nabla \xi_4) (\nabla \xi_1, \nabla \xi_3) \, dx \\ \notag &+ p(p-2)(p-4) \int_\Omega |\nabla u|^{p-6} (\nabla u, \nabla \xi_3) (\nabla u, \nabla \xi_4) (\nabla \xi_1, \nabla \xi_2) \, dx \\ \notag &+ p(p-2) \int_\Omega |\nabla u|^{p-4} (\nabla \xi_1, \nabla \xi_2) (\nabla \xi_3, \nabla \xi_4) \, dx \\ \notag &+ p(p-2) \int_\Omega |\nabla u|^{p-4} (\nabla \xi_1, \nabla \xi_3) (\nabla \xi_2, \nabla \xi_4) \, dx \\ \notag &+ p(p-2) \int_\Omega |\nabla u|^{p-4} (\nabla \xi_1, \nabla \xi_4) (\nabla \xi_2, \nabla \xi_3) \, dx. \end{align}

where $\xi_4 \in W^{1,p}(\Omega)$.

Visually, it could be easier to present this result as an $n$-th directional derivative of the $p$-th power of the norm of a vector. Namely, if $A, B_i \in \mathbb{R}^N$, then for any natural $n \geq 1$, we have

\begin{align} \notag &D^n \left(|A|^p\right) (B_1, \dots, B_n) \newline \notag &= \sum\limits_{i=0}^{\lfloor \frac{n}{2} \rfloor} |A|^{p-2(n-i)} \prod\limits_{j=0}^{n-i-1} (p-2j) \left[\sum\limits_{\sigma \in B(n,n-2i)} \prod\limits_{k=1}^{n-2i} (A, B_{\sigma(k)}) \left(\sum\limits_{\omega \in P(n,\sigma)} \prod\limits_{l=1}^{i} (B_{\omega(l,1)}, B_{\omega(l,2)}) \right) \right]. \end{align}

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