Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-rigidity of the absolutely continuous part of $\mathcal{A}$-free measures

Luigi De Masi, Carlo Gasparetto

TL;DR

This work extends Alberti’s $L^1$-density result to a broad class of first-order differential operators by introducing the notion of $k$-balanceability for $\, ext{A}$-free measures. It provides a sharp scalar theory: a scalar operator with rank $r$ is $k$-balanceable for $k\ge d+1-r$, with explicit obstruction when $k\le d-r$ and in particular non-balanceability for rank $1$; the singular part of balanced measures exhibits a precise dimensional constraint. For vector-valued operators, a structural Condition ef{hyp:solving_2} yields $(d-1)$-balanceability, and the exterior derivative on differential forms is a key instance recovering Alberti’s theorem for $1$-forms. The paper also proves a Lusin-type approximation property under Condition ef{hyp:solving_2}, and provides extensive examples and connections to the wave cone to illuminate rigidity versus nonrigidity phenomena in the absolutely continuous part of $\, ext{A}$-free measures.

Abstract

We generalize a result by Alberti, showing that, if a first-order linear differential operator $\mathcal{A}$ belongs to a certain class, then any $L^1$ function is the absolutely continuous part of a measure $μ$ satisfying $\mathcal{A}μ=0$. When $\mathcal{A}$ is scalar valued, we provide a necessary and sufficient condition for the above property to hold true and we prove dimensional estimates on the singular part of $μ$. Finally, we show that operators in the above class satisfy a Lusin-type property.

Non-rigidity of the absolutely continuous part of $\mathcal{A}$-free measures

TL;DR

This work extends Alberti’s -density result to a broad class of first-order differential operators by introducing the notion of -balanceability for -free measures. It provides a sharp scalar theory: a scalar operator with rank is -balanceable for , with explicit obstruction when and in particular non-balanceability for rank ; the singular part of balanced measures exhibits a precise dimensional constraint. For vector-valued operators, a structural Condition ef{hyp:solving_2} yields -balanceability, and the exterior derivative on differential forms is a key instance recovering Alberti’s theorem for -forms. The paper also proves a Lusin-type approximation property under Condition ef{hyp:solving_2}, and provides extensive examples and connections to the wave cone to illuminate rigidity versus nonrigidity phenomena in the absolutely continuous part of -free measures.

Abstract

We generalize a result by Alberti, showing that, if a first-order linear differential operator belongs to a certain class, then any function is the absolutely continuous part of a measure satisfying . When is scalar valued, we provide a necessary and sufficient condition for the above property to hold true and we prove dimensional estimates on the singular part of . Finally, we show that operators in the above class satisfy a Lusin-type property.
Paper Structure (13 sections, 15 theorems, 32 equations, 1 figure)

This paper contains 13 sections, 15 theorems, 32 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Notation and reduction of the problem
  3. Balancing scalar operators
  4. Proof of part \ref{['item:main_positive']} of Theorem \ref{['thm:main_scalar']}
  5. Proof of part \ref{['item:main_sharpness']} of Theorem \ref{['thm:main_scalar']}
  6. A sufficient condition for balancing vector-valued operators
  7. Examples and further observations on Condition \ref{['hyp:solving_2']}
  8. Exterior derivatives
  9. Comparing scalar and vector case
  10. Links with the wave cone
  11. Operators that do not satisfy Condition \ref{['hyp:solving_2']} and the divergence on symmetric matrix-valued vector fields
  12. Another operator that satisfies Condition \ref{['hyp:solving_2']}
  13. A Lusin-type property

Key Result

Theorem 1.2

Let $\mathcal{A}$ be a scalar-valued first order linear differential operator as in eq:first_order_operator_type with $r \coloneqq \mathop{\mathrm{rank}}\nolimits A>0$. The following hold: In particular, if $\mathop{\mathrm{rank}}\nolimits A=1$, then Question prb:alberti has a negative answer for any $k\in\{0,1,\dots,d-1\}$.

Figures (1)

  • Figure 1: The measure $\mu$ given by Lemma \ref{['lmm:1_rect_dive']}, where the center $q(Q)$ of $Q$, the cubes in $\mathcal{P}_1(Q)$ and their centers are represented.

Theorems & Definitions (29)

  • Theorem 1.2
  • Remark 1.3: $0$-balanceable operators
  • Remark 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Definition 2.1: $k$-balanceable operator
  • Proposition 2.2
  • Remark 2.3
  • Lemma 2.4
  • ...and 19 more