Thursday, April 26, 2012

sometimes the derivative is an oracle, at least one-dimensionally .. (link to a preprint)

just saw this on the arxiv today, off a preprint of n. katzourakis. it's about infinite-harmonic maps from the plane into space, but it was this result that piqued my curiosity [1]:

Theorem (κatzοurakis). Suppose $\Omega \subseteq \mathbb{R}^n$ is open and contractible and $u : \Omega \to \mathbb{R}^N$ is in $C^2(\Omega)^N$.  Then the following are equivalent:
  1. $u$ is a Rank-One map, that is ${\rm rank}(Du) \leq 1$ on $\Omega$, or equivalently there exist $\xi: \Omega \to \mathbb{R}^N$ and $w : \Omega \to \mathbb{R}^n$ such that $Du = \xi \otimes w$;
  2. there exists $f \in C^2(\Omega)$, a partition $\{B_i\}_{i=1}^\infty$ of $\Omega$ of Borel sets, where each $B_i$ equals a connected open set with a boundary portion and Lipschitz curves $\{\nu^i\}_{i=1}^\infty$ in $W^{1,\infty}_{\rm loc}(\mathbb{R})^N$ such that on each $B_i$, $u$ equals the composition of the curve $\nu^i$ with the scalar function $f$:
    $$u \;=\; \nu^i \circ f, \hspace{.5in} \text{ on } B_i \subseteq \Omega$$Moreover, $|\dot{\nu}^i| \equiv 1$ on $f(B_i)$, $\dot{\nu}^i \equiv 0$ on $\mathbb{R} \setminus f(B_i)$, and there exist $({\nu}^i)''$ on $f(B_i)$, interpreted as $1$-sides on $\partial f(B_i)$, if any.  Also,
    $$ Du \;=\; (\dot{\nu}^i \circ f) \otimes Df, \hspace{.5in} \text{ on } B_i \subseteq \Omega$$and the image $u(\Omega)$ is a $1$-rectifiable subset of $\mathbb{R}^N$.

roughly speaking, if the derivative tells you that a smooth mapping has 1-dimensional behavior, then you can actually cut it up into a single function that mimicks the mapping's behavior through curves.

(i haven't read the proof, but my guess is that the hard work is somehow done through the partitioning.  i wonder if there is a Rank-$M$ version of this result, for $M \in \mathbb{N}$ ..)
there's also a version purely for maps with components in $W^{1,\infty}(\Omega)^N$, in the same paper, but with an $L^\infty$-approximation condition via smooth maps.

[1] the blue text was added today (1 May 2012). i could swear that it was there before, but it seemed to have disappeared after reloading the blog page.  also, i decided to indent some of the last few paragraphs to highlight my guess on the subject.

No comments: