Timeline for Meaning of Alberti rank-one theorem
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| Apr 11, 2019 at 18:45 | vote | accept | CommunityBot | ||
| S Dec 19, 2018 at 20:03 | history | bounty ended | CommunityBot | ||
| S Dec 19, 2018 at 20:03 | history | notice removed | CommunityBot | ||
| Dec 11, 2018 at 20:00 | comment | added | Denis Serre | @Rene. It is hard to develop an answer to this question within a comment. I suggest that you have a look at the introduction of De Philippis & Rindler's paper. In the case covered by Alberti, the wave cone is that of rank-one matrices $a\otimes b$ with $a\cdot b=0$. | |
| Dec 11, 2018 at 19:49 | comment | added | user123457 | @DenisSerre Thank you. Heuristically, what does it mean that the singular part of the derivative of a BV function has almost everywhere values in the wave cone of the row-wise curl? | |
| Dec 11, 2018 at 19:37 | comment | added | Denis Serre | Just to mention that Alberti's result has been generalized by G. De Philippis & F. Rindler in Ann. of Math. 184 (2016), pp 1017-1039. They prove that if a vector-valued bounded measure $\mu$ satisfies a homogeneous PDE $A\mu=0$, where $A$ has constant coefficients, then the singular part $\mu^s$ takes $|\mu|^s$-almost everywhere values in the wave cone of $A$. Alberti's case occurs when $\mu$ is matrix-valued and $A$ is the row-wise curl. | |
| S Dec 11, 2018 at 18:54 | history | bounty started | CommunityBot | ||
| S Dec 11, 2018 at 18:54 | history | notice added | user123457 | Improve details | |
| S Nov 30, 2018 at 17:02 | history | bounty ended | CommunityBot | ||
| S Nov 30, 2018 at 17:02 | history | notice removed | CommunityBot | ||
| Nov 23, 2018 at 22:22 | answer | added | Bazin | timeline score: 2 | |
| Nov 23, 2018 at 17:37 | history | edited | user123457 | CC BY-SA 4.0 |
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| Nov 23, 2018 at 15:46 | comment | added | user123457 | @Mizar Thank you. Do you mind elaborating a bit in an answer on that remark? That is, what do you mean when you say "if you blow up your vector field at a.e. point with respect to this singular measure (rescaling it suitably in domain and target) what you see in the limit is a vector field depending only on, say, the first component, up to rotations"? | |
| Nov 22, 2018 at 22:17 | comment | added | Mizar | Basically, it tells you that the singular part of the (distributional) differential, which is a nasty matrix valued measure, behaves as in the simple case of "jump points" (see Definition 3.67 in the book by Ambrosio, Fusco and Pallara): namely, if you blow up your vector field at a.e. point with respect to this singular measure (rescaling it suitably in domain and target) what you see in the limit is a vector field depending only on, say, the first component, up to rotations. | |
| S Nov 22, 2018 at 15:39 | history | bounty started | CommunityBot | ||
| S Nov 22, 2018 at 15:39 | history | notice added | user123456 | Authoritative reference needed | |
| Nov 15, 2018 at 21:13 | history | asked | user123457 | CC BY-SA 4.0 |