I'm just looking for some quick and dirty intuition(and/or reading material) about the following:

I read that Hodge duality provides a way to interchange the p-Laplacian $ \Delta_p = \nabla\cdot( |\nabla|^{(p-2)}\nabla)$ and the q-Laplacian, where $1/p + 1/q = 1$. Can somebody elaborate on how this connection is made or provide a reference?

Edit: I guess I should've mentioned I'm mostly interested in connections related to the spectrum, i.e. can I say something about the spectrum of $\Delta_p$ based on that of $\Delta_q$.

Edit2: I should add that I found the reference, but it's out of my field so really I'm looking for someone to 'dumb it down' for me. The answer to this question contains the reference.

  • $\begingroup$ Any chance you could say where you read that? $\endgroup$
    – Deane Yang
    Sep 11 '15 at 0:40
  • $\begingroup$ I was actually looking for the reference. Found it! It was from a related question on p-Laplacians. I'm reading that now, but I would still appreciate some intuition. $\endgroup$
    – funda
    Sep 11 '15 at 2:32

In the simpler context of applying the $p$-Laplacian $\Delta_{p}$ to scalar functions $u \, \colon \, \Omega \subset \mathbb{R}^{N} \to \mathbb{R}$, some calculations reveal connections with complementary averaging operators for different values of $p$. Working formally, for smooth $u$, $1 < p < \infty$, and $p^{-1} + q^{-1} = 1$, $$\Delta_{p} u ~ := ~ \operatorname{div}{ \left( | \nabla u |^{p-2} \nabla u \right) } ~ = ~ p | \nabla u |^{p-2} \left( p^{-1} \Delta_{1} u + q^{-1} \Delta_{\infty} u \right) \ , $$ where $$ \Delta_{\infty} u ~ := ~ | \nabla u |^{-2} \sum_{i,j}{ u_{i}u_{j} u_{ij} } \quad \mbox{and} \quad \Delta_{1} u ~ := ~ \Delta u - \Delta_{\infty} u$$ are, respectively, the $\infty$-Laplacian and the $1$-Laplacian. (See, e.g., Peres & Sheffield.)

Now note that $\Delta_\infty u(x)$ measures the difference between $u(x)$ and its midrange on a ball (or sphere) centered at $x$, $$-\Delta_{\infty} u (x) ~ \propto ~ \lim_{h \to 0}{ \frac{2}{h^2} \left( u(x) - \operatorname{midrange}_{B(x,h)}{ u } \right) } \ , $$ where $$\operatorname{midrange}_{B(x,h)}{ u } ~ := ~ \frac{1}{2} \, \left( \max_{\overline{B(x,h)}}{ u } + \min_{\overline{B(x,h)}}{ u } \right) \ ,$$ while $\Delta_{1} u(x)$ measures the difference between $u(x)$ and its median over a ball (or sphere) centered at $x$, $$-\Delta_{1} u(x) ~ \propto ~ \lim_{h \to 0}{ \frac{2}{h^2} \left( u(x) - \operatorname{median}_{\overline{B(x,h)}}{ u } \right) } \ , $$ where the median $m$ of $u$ over $\overline{B(x,h)}$ is the number such that $$ | \left\{ u < m \right\} | \leq \frac{1}{2} |B(x,h)| \quad \mbox{and} \quad | \left\{ u > m \right\} | \leq \frac{1}{2} |B(x,h)| \ .$$ (See, e.g., Kawohl, Manfredi & Parviainen and Rudd.)

Putting all of this together shows that the $p$-Laplacian of $u$ compares $u$ to a linear combination of its neighboring average and midrange when $p > 2$, while it compares $u$ to a linear combination of its neighboring average and median when $1 < p < 2$. These formal calculations can be made rigorous in the context of viscosity solutions.

  • $\begingroup$ Thanks for your quick answer! This is a form of intuition, but not the kind I'm looking for unfortunately, since it doesn't really relate $\Delta_p$ to $\Delta_q$ but instead tries to explain $\Delta_p$ itself, in the different regimes. I must confess the reference I posted in my second edit is way out of my field, so I guess I'm looking for someone to dumb it down. Also as I mentioned in my first edit, I'm looking for connections relating their spectra. $\endgroup$
    – funda
    Sep 11 '15 at 13:17

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