Friday, April 13, 2012

in which i suggest an "N" instead of a "p" ..

while browsing the title/abstract of a preprint of cheη, pοnnusamy, and waηg, i read this excerpt ..
"In this paper, we investigate the properties of $p$-harmonic mappings in the unit disk $|z| \leq 1$. First, we discuss the convexity, the starlikeness and the region of variability of some classes of $p$-harmonic mappings."
.. and became excited:

new geometric results about $p$-harmonic mappings?

so i clicked on the PDF and read the first page ..

yeah, yeah, $p$-harmonic functions solve $\Delta^pf = 0$  ..
wait: why should $f$ be be $C^{2p}$-smooth?

and then i realised that i mistook a superscript for a subscript.  you see, these authors mean $p$ as an exponent for composition,
$$\Delta^pf \;=\; (\underbrace{\Delta \circ \cdots \circ \Delta}_\textrm{ $p$ times })f$$
whereas the $p$-Laplacian that i know and love from the literature is a nonlinear operator:
$$\Delta_pf \;:=\; \operatorname{div}[|\nabla f|^{p-2}\nabla f].$$

either i need more sleep or more coffee, today.

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