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Accepted Manuscript Adolfo Arroyo-Rabasa An elementary approach to the dimension of measures satisfying a first-order linear PDE constraint Proceedings of the American Mathematical Society DOI: 10.1090/proc/14732 Accepted Manuscript This is a preliminary PDF of the author-produced manuscript that has been peer-reviewed and accepted for publication. It has not been copyedited, proofread, or finalized by AMS Production staff. Once the accepted manuscript has been copyedited, proofread, and finalized by AMS Production staff, the article will be published in electronic form as a \Recently Published Article" before being placed in an issue. That electronically published article will become the Version of Record. This preliminary version is available to AMS members prior to publication of the Version of Record, and in limited cases it is also made accessible to everyone one year after the publication date of the Version of Record. The Version of Record is accessible to everyone five years after publication in an issue. AN ELEMENTARY APPROACH TO THE DIMENSION OF MEASURES SATISFYING A FIRST-ORDER LINEAR PDE CONSTRAINT ADOLFO ARROYO-RABASA Abstract. We give a simple criterion on the set of probability tangent mea- sures Tan(µ, x) of a positive Radon measure µ, which yields lower bounds on the Hausdorff dimension of µ. As an application, we give an elementary and purely algebraic proof of the sharp Hausdorff dimension lower bounds for first-order linear PDE-constrained measures; bounds for closed (measure) dif- ferential forms and normal currents are further discussed. A weak structure theorem in the spirit of [Math. 184(3) (2016), pp. 1017{1039] is also discussed for such measures. 1. Introduction The question of determining the dimension of a vector-valued Radon measure satisfying a PDE-constraint is a longstanding one. A good starting point is the understanding of curl-free (gradient) measure fields and their dimensional prop- erties. From a historical perspective, the seminal work of De Giorgi [9] on the structure of sets of finite perimeter and the co-area formula [13] of Fleming & Rishel allowed for a transparent proof of the estimate jDuj Hd−1, for all dis- tributional gradients Du represented by a Radon measure. Later on, the work of Federer [12, Section 4.1.21] , among many others, sparked new groundbreaking ideas in geometric measure theory and successfully extended this result to the es- m m d 1 timate kT k I H , for all m-dimensional normal currents T 2 Nm(R ). Motivated by new challenges, only recently these results have been further extended to deal with more general differential constraints in the context of A-free measures. ` k ` k Namely, in [6, Corollary 1.4] it has been shown that jµj I P H P for vector- valued measures µ satisfying a generic constraint of the form P (D)µ = 0, where P (D) is a kth-order linear partial differential operator with constant coefficients k and ` k is a positive integer depending only on the principal symbol of P (D); P P see also [7, 19]. This ` k dimensional estimate turns out to be sharp for first-order P ` k operators. However, determining whether jµj H P is optimal for operators of order k > 1 seems to be a delicate question (see [6, Conjecture 1.6]; similar and related questions have been previously advanced in [19] and [18, Question 5.11]). The compendium of results mentioned above possesses a strong geometrical fla- vor which, for instance, addresses several rectifiability properties; here, we shall focus on the simpler task of discussing Hausdorff dimension bounds for such mea- sures. However stronger, the aforementioned results rely on a significantly deeper Date: April 15, 2019. 2010 Mathematics Subject Classification. Primary 28A78, 49Q15; Secondary 35F35. Key words and phrases. Hausdorff dimension, A-free measure, PDE constraint, tangent mea- sure, structure theorem, normal current. This project has received funding from the European Research Council (ERC) under the Eu- ropean Union's Horizon 2020 research and innovation programme, grant agreement No 757254 (SINGULARITY). 1 m d Here, I is the m-integral-geometric measure on R . 1 This15 isApr a pre-publication 2019 09:55:30 version of thisEDT article, which may differ from the final published version. CopyrightAnalysis restrictions may apply. Version 2 - Submitted to Proc. Amer. Math. Soc. 2 A. ARROYO-RABASA and more technical machinery. Instead, the purpose of this work is to present a different and didactic method to address (only) dimensional bounds. We give a self-contained and elementary proof of the Hausdorff dimension (sharp) bounds for measures µ 2 M(Ω;E) solving (in the sense of the distributions) a first-order equation of the form d X ∗ (1) P (D)µ := Pi[@iµ] + P0µ = 0 ;P0;Pi 2 F ⊗ E ; i=1 where E; F are finite dimensional euclidean spaces. The cornerstone of our proof rests on a rather simple invariance criterion, affect- ing all normalized blow-ups of a given positive Radon measure σ, which effortlessly yields a lower bound on the Hausdorff dimension dimH(σ); where, as usual s dimH(σ) := sup 0 ≤ s ≤ d : σ H : This criterion (contained in Lemma 9) links the vector-space dimension, of those directions with respect to which a blow-up of σ may be an invariant measure, to a lower bound of the Hausdorff dimension. In particular, re-directing the study of dimensional estimates for measures satisfying (1), to the study of the struc- tural rigidity of their sets Tan(jµj; x) of probability tangent measures (described in Section 3).2 The advantage of this viewpoint lies in the fact that the principal symbol d X d ξ 7! P(ξ) := ξiPi; ξ 2 R ; i=1 being linear as function of ξ, precisely characterizes those directions where tangent measures are invariant measures. Therefore, allowing one to define a notion of dimension associated to the principal part of the operator by setting ? (2) ` := min dim fP[e] ≡ 0g ; P e2Enf0g where we have used the short-hand notation fP[e] ≡ 0g := f ξ : P(ξ)[e] = 0 g. It is worthwhile to mention that this definition of dimension agrees with the definition given in [6, Equation (1.6)]. Let us also point out that, in the context of cocanceling operators (see [20]; see also [6, 18]), P (D) is an `P-cocanceling operator. Our main result is contained in the following theorem: Theorem 1. Let Ω ⊂ Rd be an open set, let P (D) be a first-order differential operator as in (1), and let µ 2 M(Ω; E) be a solution of the equation P (D)µ = 0 in the sense of distributions on Ω: Then, dimH(jµj) ≥ `P: Moreover, this estimate is sharp since the measure ` ? µ = e H P xfP[e] ≡ 0g is an homogeneous solution of (1) on Rd, whenever e 2 E is any vector at which the minimum in (2) is attained. Remark 2. The proof of Theorem 1 does not require, in any way, the structure theorem for PDE-constrained measures [11, Theorem 1.1]. 2A similar method for establishing dimensional estimates was developed by Ambrosio & Soner in [4]. The same method has been also discussed in [15] using the slightly more restrictive notion d Tσ(x) ⊂ R of tangent space to a measure σ, as introduced by Bouchitte´, Buttazzo and Seppecher in [8]. This15 isApr a pre-publication 2019 09:55:30 version of thisEDT article, which may differ from the final published version. CopyrightAnalysis restrictions may apply. Version 2 - Submitted to Proc. Amer. Math. Soc. AN ELEMENTARY APPROACH TO THE DIMENSION OF MEASURES 3 dµ At all points x where [P (D)◦ djµj (x)] is elliptic, that is, precisely when the polar dµ djµj (x) does not belong to the wave cone set [ ΛP := ker P(ξ) ⊂ E; d ξ2R nf0g the sets Tan(jµj; x) turn out to be trivial (containing only fully-invariant measures). The invariance criterion then allows us to give the following soft version of [11, Theorem 1.1] (see also [1] in the case of gradients): Corollary 3 (weak structure theorem). Let Ω ⊂ Rd be an open set, let P (D) be a first-order differential operator as in (1), and let µ 2 M(Ω; E) be a solution of the equation P (D)µ = 0 in the sense of distributions on Ω: Then, dµ s jµjx x 2 Ω : djµj (x) 2= ΛP H for all 0 ≤ s < d: Remark 4. The results contained in Theorem 1 and Corollary 3 apply to solutions of the inhomogeneous equation P (D)µ = τ 2 M(Ω; F ): To see this, letµ ~ = (µ, τ), E~ = E × F , and consider the operator P~(D)~µ = P (D)µ − τ. Further comments. Both Theorem 1 and Lemma 9 do not lead to rectifiability ` ` properties of jµj, nor estimates of the form jµj I P (or even jµj H P ) by the methods presented in this note. This assertion is in line with the following observation: the shortcoming of Corollary 3 |with respect to the (strong) structure theorem| lies in the requirement of s being strictly smaller than d. As it has been remarked by De Lellis (see [10, Proposition 3.3]), Preiss' example [17, Example 5.8(1)] of a purely singular measure with only trivial tangent measures hinders the hope for a traditional blow-up strategy leading to the estimate in the critical case s = d.3 In forthcoming work [5], it will be shown that all functions u :Ω ! Rd of bounded deformation satisfy the following rigidity property: every probability tangent mea- sure τ 2 Tan(Eu; x) can be split as a sum of plane-wave measures (here, Eu = 1 t d d 2 (Du+Du ) 2 M(Ω; sym(R ⊗R )) is the distributional symmetric gradient of u).
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