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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 \OmegaMoreover, |\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 \Omegaand the image u(\Omega) is a 1-rectifiable subset of \mathbb{R}^N.


interesting:
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.

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