Heuristically what does Alberti's rank-one theorem imply about the structure of a $\mathrm{BV}$ vector field $\boldsymbol{b}$?

Is it rigorously fair to say that the level lines of $\boldsymbol{b}$ are all "parallel" and pointing in one direction? Why?

This question has an open bounty worth +50 reputation from Rene ending in 5 days.

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  • 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. – Mizar Nov 22 at 22:17
  • @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"? – Rene Nov 23 at 15:46
  • 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. – Denis Serre yesterday
  • @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? – Rene yesterday
  • @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 $a\cdot b=0$. – Denis Serre yesterday

Let $X=\sum_ja_j(x)\frac{\partial }{\partial x_j}$ be a $BV$ vector field in an open subset of $\mathbb R^n.$ Alberti's theorem says that $$ DX_s=(\frac{\partial a_j }{\partial x_k})_{1\le j,k\le n}=S \otimes \eta, \quad \text{$S(x)$ tangent vector at $x$, $\eta(x)$ cotangent vector at $x$} $$ i.e for $T$ tangent at $x$, $ (DX)_s(x) T= \langle \eta(x), T\rangle S(x). $ Intuitively, it means essentially that, near each point, you can find a coordinate system such that each coefficient $a_j$ of your vector field is in fact a function $$ a_j(x', x_n) \text{ such that}\ \frac{\partial a_j}{\partial x'} \in L^1,\quad \frac{\partial a_j}{\partial x_n} \text{is a Radon measure}, $$ where $x'$ is $n-1$ dimensional and $x_n$ has just one dimension.

Let me add a couple of comments. You have a canonical decomposition of $DX$. What matters is the singular part. When I write as above $DX_s=S\otimes \eta$, it means that for a tangent vector $T$, $$\underbrace{DX_s}_{\text{matrix}}T=\langle\eta,T\rangle \underbrace{S}_{\text{vector}}.$$ You may decide that $\eta=e_n^*$ so that $DX_sT=T_n S$ and if you know that the divergence of $X$ is absolutely continuous, you have $S_n=0$, so that you may assume that $S=e_1$. As a result, $$ DX_s e_n=e_1,\quad DX_s e_k=0 \text{ for $1\le k\le n-1$}. $$ Since $DX=\bigl(\frac{\partial a_j}{\partial x_k}\bigr)$, $j$ line-index, $k$ column-index, you get that only the last column $$ \frac{\partial a_j}{\partial x_n}\quad \text{can be a Radon measure},\quad \frac{\partial a_j}{\partial x_k}, k\le n-1\quad \text{(any $j$) is $L^1$} $$ With our assumption on the divergence and our choice $S=e_1$, we have only $$ \frac{\partial a_1}{\partial x_n} \text{singular, other entries $L^1$.} $$ Of course, you have to keep in mind that $S,\eta$ depend on the point $x$, but through an approximation procedure, you may indeed assume that singularity occurs by differentiating only in one direction (and with null divergence (or ac) is concerning only one off-diagonal coefficient).

  • Hi. I've opened a bounty because I'd like to see more details on this topic. In particular, why does Alberti's theorem imply that locally singularities of BV functions are necessarily one-directional and that after a blow-up at $|D^{sing} X|$-almost every point, the blow-up limit measure depends only on a single direction. – Rene 2 days ago
  • Thank you. The additional remarks are very useful. Could you comment on what you mean precisely when you say "through an approximation procedure, you may indeed assume that singularity occurs by differentiating only in one direction (and with null divergence (or ac) is concerning only one off-diagonal coefficient)."? – Rene yesterday
  • Also, by chance do you have a reference at hand that gives some heuristic insight into Alberti theorem? – Rene yesterday
  • I have no more heuristic explanations. If you want to enter the details, I would recommend the paper MR2124585,Transport equations with partially BV velocities, by N. Lerner and the L. Ambrosio article quoted there. – Bazin yesterday

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