while browsing the title/abstract of a preprint of cheη, pοnnusamy, and waηg, i read this excerpt ..

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

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.

"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?**awesome!*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:

$$\Delta_pf \;:=\; \operatorname{div}[|\nabla f|^{p-2}\nabla f].$$

*[sighs]*either i need more sleep or more coffee, today.

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