Dimensional estimates and rectifiability for measures satisfying linear PDE constraints
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Abstract
We establish the rectifiability of measures satisfying a linear PDE constraint. The obtained rectifiability dimensions are optimal for many usual PDE operators, including all firstorder systems and all secondorder scalar operators. In particular, our general theorem provides a new proof of the rectifiability results for functions of bounded variations (BV) and functions of bounded deformation (BD). For divergencefree tensors we obtain refinements and new proofs of several known results on the rectifiability of varifolds and defect measures.
Keywords and phrases
Rectifiability dimensional estimate \(\mathcal {A}\)free measure PDE constraint1 Introduction
In [DR16] it was shown that for any \(\mathcal {A}\)free measure there is a strong constraint on the directions of the polar at singular points:
Theorem 1.1

If \(\mathcal {A}=\mathrm{curl}\), the above theorem gives a new proof of Alberti’s rankone theorem [Alb93] (see also [MV16] for a different proof based on a geometrical argument).

If \(\mathcal {A}={{\,\mathrm{div}\,}}\), combining Theorem 1.1 with the result of [AM16], one obtains the weak converse of Rademacher’s theorem (see [DMR17, GP16, KM18] for other consequences in metric geometry).
Definition 1.2
Our main result establishes that the polar of an \(\mathcal {A}\)free measure is constrained to lie in a smaller cone on \(\mathcal {I}^\ell \)null sets:
Theorem 1.3
Note that, by taking \(\ell =d\), Theorem 1.3 recovers Theorem 1.1. As a corollary we obtain the following dimensional estimates on \(\mathcal {A}\)free measures; see also [Arr18] for a different proof of (1.4) in the case of firstorder systems.
Corollary 1.4
Theorem 1.5
Theorem 1.5 contains the classical rectfiability result for the jump part of the gradient of a \(\mathrm{BV}\) function, see [AFP00], and the analogous result for \(\text {BD}\), see [Koh79, ACD97]. By choosing \(\mathcal {A}={{\,\mathrm{div}\,}}\) we also recover and (in some cases slightly generalize) several known rectifiability criteria, such as Allard’s rectifiability theorem for varifolds [All72], its recent extensions to anisotropic energies [DDG18], the rectifiability of generalized varifolds established in [AS97], and the rectifiability of various defect measures in the spirit of [Lin99], see also [Mos03]. We refer the reader to Sect. 3 for some of these statements.
 By some measuretheoretic arguments one shows that, up to a subsequence,for some positive measure \(\sigma \) and some fixed vector \(\lambda \).$$\begin{aligned} r^{\ell } T^{x_0,r}\mu \overset{*}{\rightharpoonup }\lambda \sigma \end{aligned}$$

One exploits this information together with the \(\mathcal {A}^k\)freeness of \(\lambda \sigma \), where \(\mathcal {A}^k\) is the principal part of \(\mathcal {A}\), to deduce that \(\sigma \) is translationinvariant along the directions in an \(\ell \)dimensional plane \(\pi \) and thus Open image in new window . In this step one usually uses that \(\pi \) is uniquely determined by \(\lambda \) and \(\mathcal {A}\).
Let us now briefly discuss the optimality of our results. First note that (1.6) and (1.7) are true whenever an \(\mathcal {A}\)free measure \(\mu \) has a nontrivial part concentrated on an \(\ell \)rectifiable set R, see Lemma 2.3 below.
Conjecture 1.6
We note that the same conjecture has also been advanced by Raita in [Rai18, Question 5.11]; also see [AW17, Conjecture 1.5].
We conclude this introduction by remarking that the above results can be used to provide dimensional estimates and rectifiability results for measures whose decomposability bundle, defined in [AM16], has dimension at least \(\ell \). Namely, in this case the measure is absolutely continuous with respect to \(\mathcal {I}^{\ell }\) and the set where the upper \(\ell \)dimensional density is positive, is rectifiable, compare with [Bat17, Theorem 2.19] and with [AMS]. However, since by its very definition the dimension of the decomposability bundle is stable under projections, in this setting one can directly rely on [DR16, Corollary 1.12]. This is essentially the strategy followed in the cited references.
2 Proofs
The proof of Theorem 1.3 is a combination of ideas from [DR16] and [DDG18]. We start with the following lemma.
Lemma 2.1
 (a1)\(\mathcal {B}\lambda \) is elliptic for some \(\lambda \in \mathbb {R}^m\), that is,where \(\mathbb {B}(\xi ):=\sum _{\beta =k} A_{\beta } \xi ^\beta \in \mathbb {R}^n \otimes \mathbb {R}^m\);$$\begin{aligned} \lambda \notin \ker \mathbb {B}(\xi ) \quad \text {for all }\xi \in \mathbb {R}^\ell \setminus \{0\}, \end{aligned}$$
 (a2)
\(\{({{\,\mathrm{\mathrm {Id}}\,}}\Delta )^{\frac{s}{2}}\mathcal {B}\nu _j\}_j\) is precompact in \(\mathrm {L}^1(B_1^\ell ;\mathbb {R}^n)\) for some \(s<k\);
 (a3)
\(\displaystyle \lim _{j\rightarrow \infty } \int _{B^\ell _1} \biggl  \frac{\mathrm{d}\nu _j}{\mathrm{d}\nu _j}  \lambda \biggr  \;\mathrm {d}\nu _j= 0\).
Proof
The proof is a straightforward modification of the main step of the proof of [DR16, Theorem 1.1], see also [All86] and [Rin18, Chapter 10]. We give it here in terse form for the sake of completeness.
 (i)
the symbol for \(T_0\) is a Hörmander–Mihlin multiplier (i.e. a pseudodifferential operator with smooth symbol of order 0) since, due to (a1), \({\mathbb {B}}(\xi )\lambda  \ge c\xi ^k\) for some \(c > 0\) and all \(\xi \in \mathbb {R}^\ell \);
 (ii)
\(T_1\) is a pseudodifferential operator with smooth symbol of order \(k\);
 (iii)
\(T_2\) is a pseudodifferential operator with smooth symbol of order \(1\);
 (iv)
\(T_3\) is a pseudodifferential operator with smooth symbol of order \(2k\).
 (I)\(T_0\) is bounded from \(\mathrm {L}^1\) to \(\mathrm {L}^{1, \infty }\) (weak\(\mathrm {L}^1\)), see e.g. [Gra14, Theorem 6.2.7]. Owing to (a3), it follows that for \(j \rightarrow \infty \) we obtainThus,$$\begin{aligned} \int V_j \;\mathrm {d}x&\le \int \chi \, \varphi _{\epsilon _j}\!\star \!\left( \biggl  \frac{\mathrm {d}\nu _j}{\mathrm {d}\nu _j}  \lambda \biggr  \, \nu _j\right) \;\mathrm {d}x \\&\le \int _{B_1}\biggl  \frac{\mathrm {d}\nu _j}{\mathrm {d}\nu _j}  \lambda \biggr  \;\mathrm {d}\nu _j \\&\rightarrow 0. \end{aligned}$$That is, \(T_0[V_j] \rightarrow 0\) in measure.$$\begin{aligned} \sup _{t \ge 0} t \, \mathcal {L}^d(\{ T_0[V_j]>t\}) \le C \int V_j \;\mathrm {d}x \rightarrow 0 \qquad \text {as }j \rightarrow \infty . \end{aligned}$$
 (II)
Due to (a2), \(T_1[f_j]\) is precompact in \(\mathrm {L}^1\) (this follows directly by the symbolic calculus [Ste93, Section VI.3] or direct manipulation of Fourier multipliers).
 (III)
\(T_2\) and \(T_3\) are compact operators from \(\mathrm {L}^1_c\) to \(\mathrm {L}^1_{\text {loc}}\) (see for instance [Ste93, Propositions VI.4, VI.5] in conjunction with Lemma 10.1 in [DR16] or Lemma 10.11 in [Rin18]) and thus the families \(\{T_2[W_j]\}\), \(\{T_3[u_j]\}\) are precompact in \(\mathrm {L}^1\).
The following is Lemma 2.2 in [DR16], we report here its straightforward proof for the sake of completeness.
Lemma 2.2
 (i)
\(f_j\overset{*}{\rightharpoonup }0\) in \( \mathrm {C}^\infty _c(B_1)^*\);
 (ii)the negative parts \(f_j^ := \max \{f_j,0\}\) of the \(f_j\)’s converge to zero in measure, i.e.,$$\begin{aligned} \lim _{j\rightarrow \infty }\, \bigl \bigl \{\, x \in B_1 \ \ \mathbf : \ \ f_j^(x)> \delta \,\bigr \}\bigr  = 0 \qquad \text {for every }\delta >0; \end{aligned}$$
 (iii)
the family of negative parts \(\{f_j^{}\}\) is equiintegrable.
Proof
Proof of Theorem 1.3
 (b1)\(\lambda :=\dfrac{\mathrm{d}\mu }{\mathrm{d}\mu }(x_0)\) exists, belongs to \({\mathbb {S}}^{m1}\), and satisfies$$\begin{aligned} \mathbb {A}^k(\xi )\lambda \ne 0 \qquad \text {for all }\xi \in \pi _0\setminus \{0\}; \end{aligned}$$(2.3)
 (b2)
 (b3)forthe following convergence holds:$$\begin{aligned} \mu _j:=\frac{T^{x_0,r_j}_\#\mu }{\mu (B_{2r_j}(x_0))} \end{aligned}$$for some \(\sigma \in \mathcal {M}^+(B_2)\) with Open image in new window . Here, \(T^{x_0,r_j}(x):=\dfrac{xx_0}{r_j}\).$$\begin{aligned} \mu _j:=\frac{T^{x_0,r_j}_\#\mu }{\mu (B_{2r_j}(x_0))}\overset{*}{\rightharpoonup } \sigma \end{aligned}$$
Before proving Theorem 1.5, let us start with the following elementary lemma:
Lemma 2.3
Proof
\(\square \)
Proof of Theorem 1.5
3 Applications
In this section we sketch applications of the abstract results to several common differential operators \(\mathcal {A}\). In this way we recover and improve several known results.
3.1 Rectifiability of \(\text {BV}\)gradients.
3.2 Rectifiability of symmetrized gradients.
3.3 Rectifiability of varifolds and defect measures.
Proposition 3.1
Proof
 (a)
One obtains that Open image in new window is rectifiable while in [DDG18] only the rectifiability of Open image in new window is shown (here, \(\theta _{*,\ell }\) is the lower \(\ell \)dimensional Hausdorff density map).
 (b)
If one only wants to get the rectifiability of the measure Open image in new window , then condition (i) in [DDG18, Definition 1.1] is enough. This allows, in the case \(\ell =d1\), to work with convex but not necessarily strictly convex integrands.
Notes
Acknowledgements
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, Grant Agreement No. 757254 (SINGULARITY), and from the INDAMgrant “Geometric Variational Problems”. We thank the anonymous referee for various comments that improved the presentation of this paper.
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