Another simple proof of the 1-dimensional flat chain conjecture
We provide a short proof of the 1-dimensional flat chain conjecture.
math.MG↗
arXiv subjects
Thierry De Pauw
Publications and source records attributed to Thierry De Pauw.
On divergence operators: Free space and vanishing charges
We use localized topologies to prove existence and optimal regularity results for the divergence equation $\mathrm{div} (v) = F$ in critical cases $v \in L_1(Ω;\mathbb{R}^m)$ or $v \in C_0(Ω;\mathbb{R}^m)$, i.e. we characterize those $F$ for which a solution $v$ exists whose norm is bounded by an appropriate norm of $F$. We assume $Ω$ satisfies a Poincaré inequality or an extension property. We apply the general theory to give examples of admissible $F$ in each case.
math.AP↗
A brief review of Radon-Nikodýmification
We review recent results on Radon-Nikodýmification of abstract measure spaces, the particular case of integral geometric measure, and applications to the dual of SBV.
math.FA↗
Localized locally convex topologies
Motivated by ill-posed PDEs such as $\mathrm{div} (v) = F$ we study locally convex topologies $\mathcal{T}_{\mathcal{C}}$ on real vector spaces $X$ that are a ``localized'' version of a locally convex topology $\mathcal{T}$ to members of a family $\mathcal{C}$ of convex subsets of $X$. The distributions $F$ arising as $\mathrm{div} (v)$ are expected to be the members of the dual of well-chosen $X$ with respect to an appropriate localized topology $\mathcal{T}_{\mathcal{C}}$. In this work, the emphasis is on studying the functional analytic properties of $\mathcal{T}_{\mathcal{C}}$, according to those of $\mathcal{T}$ and $\mathcal{C}$. For instance, we show that in all foreseen applications, $\mathcal{T}_{\mathcal{C}}$ is sequential but none of Fréchet-Urysohn, barrelled, and bornological. These awkward phenomena are illustrated explicitly on a specific example corresponding to the distributional divergence of continuous vector fields in $\mathbb{R}^m$. We also show that, essentially, $\mathcal{T}_{\mathcal{C}}$ is semireflexive if and only if members of $\mathcal{C}$ are $\mathcal{T}$-compact. This leads to an abstract existence theorem, thereby establishing a general scheme for characterizing those $F$ such that $\mathrm{div} (v) = F$ for various classes of regularity of $v$, various classes of domains, and various boundary conditions.
math.FA↗
Banach-Lamperti for Kurzweil-Henstock
We identify isometric isomorphisms of the space of Kurzweil-Henstock integrable functions as bi-absolutely-continuous changes of variable.
math.FA↗
Quantified compactness in Lipschitz-free spaces of $[-1,1]^n$
We show that the members of the Lipschitz-free space of $[-1,1]^n$ are exactly the 0-dimensional flat currents whose "boundary" vanishes. The connection with normal and flat currents allows to use the Federer-Fleming compactness and deformation theorems in this context. We characterize the compact subsets of this Lipschitz-free space and we quantify their $ε$-entropy.
math.FA↗
A regularity property of fractional Brownian sheets
A function $f$ defined on $[0, 1]^d$ is called strongly chargeable if there is a continuous vector-field $v$ such that $f(x_1, \dots,x_d)$ equals the flux of $v$ through the rectangle $[0, x_1] \times \cdots \times [0, x_d]$ for all $(x_1, \dots, x_d) \in [0, 1]^d$. In other words, $f$ is the primitive of the divergence of a continuous vector-field. We prove that the sample paths of the Brownian sheet with $d \geq 2$ parameters are almost surely not strongly chargeable. On the other hand, those of the fractional Brownian sheet of Hurst parameter $(H_1, \dots, H_d)$ are shown to be almost surely strongly chargeable whenever \[ \frac{H_1 + \cdots + H_d}{d} > \frac{d - 1}{d}. \]
math.PR↗
A representation formula for members of SBV dual
We give an integral representation formula for members of the dual of $SBV(\mathbb{R}^n)$ in terms of functions that are defined on $\hat{\mathbb{R}}^n$, an appropriate fiber space that we introduce, consisting of pairs $(x,[E]_x)$ where $[E]_x$ is an approximate germ of an $(n-1)$-rectifiable set $E$ at $x$.
math.CA↗
Localizable locally determined measurable spaces with negligibles
We study measurable spaces equipped with a $σ$-ideal of negligible sets. We find conditions under which they admit a localizable locally determined version -- a kind of fiber space that describes locally their directions -- defined by a universal property in an appropriate category that we introduce. These methods allow to promote each measure space $(X, \mathcal{A},μ)$ to a strictly localizable version $(\hat{X}, \hat{\mathcal{A}},\hatμ)$, so that the dual of $L_1(X, \mathcal{A}, μ)$ is $L_\infty(\hat{X},\hat{\mathcal{A}},\hatμ)$. Corresponding to this duality is a generalized Radon-Nikodým theorem. We also provide a characterization of the strictly localizable version in special cases that include integral geometric measures, when the negligibles are the purely unrectifiable sets in a given dimension.
math.CA↗
Density estimate from below in relation to a conjecture of A. Zygmund on Lipschitz differentiation
Letting $A \subset \mathbb{R}^n$ be Borel measurable and $W_0 : A \to \mathbb{G}(n,m)$ Lipschitzian, we establish that \begin{equation*} \limsup_{r \to 0^+} \frac{\mathcal{H}^m \left[ A \cap B(x,r) \cap (x+ W_0(x))\right]}{α(m)r^m} \geq \frac{1}{2^n}, \end{equation*} for $\mathcal{L}^n$-almost every $x \in A$. In particular, it follows that $A$ is $\mathcal{L}^n$-negligible if and only if $\mathcal{H}^m(A \cap (x+W_0(x))=0$, for $\mathcal{L}^n$-almost every $x \in A$.
math.CA↗
Linear isoperimetric inequality for normal and integral currents in compact subanalytic sets
The isoperimetric inequality for a smooth compact Riemannian manifold $A$ provides a positive ${\bf c}(A)$, so that for any $k+1$ dimensional integral current $S_0$ in $A$ there exists an integral current $ S$ in $A$ with $\partial S=\partial S_0$ and ${\bf M}(S)\leq {\bf c}(A){\bf M}(\partial S)^{(k+1)/k}$. Although such an inequality still holds for any compact Lipschitz neighborhood retract $A$, it may fail in case $A$ contains a single polynomial singularity. Here, replacing $(k+1)/k$ by $1$, we find that a linear inequality ${\bf M}(S)\leq {\bf c}(A){\bf M}(\partial S)$ is valid for any compact algebraic, semi-algebraic, or even subanalytic set $A$. In such a set, this linear inequality holds not only for integral currents, which have $\boldsymbol{Z}$ coefficients, but also for normal currents having $\boldsymbol{R}$ coefficients and generally for normal flat chains with coefficients in any complete normed abelian group. A relative version for a subanalytic pair $B\subset A$ is also true, and there are applications to variational and metric properties of subanalytic sets.
math.AP↗
Undecidably semilocalizable metric measure spaces
We characterize measure spaces such that the canonical map $L_\infty \to L_1^*$ is surjective. In case of $d$ dimensional Hausdorff measure of a complete separable metric space $X$ we give two equivalent conditions. One is in terms of the order completeness of a quotient Boolean algebra associated with measurable sets and with locally null sets. Another one is in terms of the possibility to decompose space in a certain way into sets of nonzero finite measure. We give examples of $X$ and $d$ so that whether these conditions are met is undecidable in ZFC, including one with $d$ equals the Hausdorff dimension of $X$.
math.FA↗
Partial regularity of almost minimizing rectifiable G chains in Hilbert space
We adapt to an infinite dimensional ambient space E.R. Reifenberg's epiperimetric inequality and a quantitative version of D. Preiss' second moments computations to establish that the set of regular points of an almost mass minimizing rectifiable $G$ chain in $\ell_2$ is dense in its support, whenever the group $G$ of coefficients is so that $\{\|g\| : g \in G \}$ is discrete and closed.
math.DG↗
On Lebesgue null sets
Letting A be a Borel subset of n dimensional Euclidean space, and W(x) be an m dimensional affine subspace containing x and varying in a Lipschitz way according to x, we establish that A is Lebesgue null if and only if $A \cap W(x)$ has m dimensional Hausdorff measure vanishing for almost every x.
math.CA↗
On the existence of mass minimizing rectifiable G chains in finite dimensional normed spaces
We introduce the notion of density contractor of dimension m in a finite dimensional normed space X. If m+1=dim X this includes the area contracting projectors on hyperplanes whose existence was established by H. Busemann. If m=2, density contractors are an ersatz for such projectors and their existence, established here, follows from work by D. Burago and S. Ivanov. Once density contractors are available, the corresponding Plateau problem admits a solution among rectifiable G chains, regardless of the group of coefficients G. This is obtained as a consequence of the lower semicontinuity of the $m$ dimensional Hausdorff mass, of which we offer two proofs. One of these is based on a new type of integral geometric measure.
math.CA↗
Existence of $p$ harmonic multiple valued maps into a separable Hilbert space
The general existence of $p$-Dirichlet energy minimizing maps into $Q_Q(l_2)$ is obtained.
math.AP↗
Multiple valued maps into a separable Hilbert space that almost minimize their $p$ Dirichlet energy or are squeeze and squash stationary
Let $f : U\subset\Rm \to \calQ_Q(\ell_2)$ be of Sobolev class $W^{1,p}$, $1 < p < \infty$. If $f$ almost minimizes its $p$ Dirichlet energy then $f$ is Hölder continuous. If $p=2$ and $f$ is squeeze and squash stationary then $f$ is in VMO.
math.AP↗