Добірка наукової літератури з теми "Rectifiable and purely unrectifiable sets"

We propose here a way to extend to all the 1-set of ℝ2 the well-known affine length which was just defined for a C2 curve. Moreover, this leads us to define the affine dimension of a 1-set which can be used for discriminate rectifiable 1-sets from unrectifiable 1-sets of ℝ2.

Abstract A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean space for all n ≥ 2 in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an L2 gauge the extent to which μ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the classical theorems of Besicovitch, Morse and Randolph, and Moore, we do not assume an a priori relationship between μ and 1-dimensional Hausdorff measure H1. We also characterize purely 1-unrectifiable Radon measures, i.e. locally finite measures that give measure zero to every finite length curve. Characterizations of this form were originally conjectured to exist by P. Jones. Along the way, we develop an L2 variant of P. Jones’ traveling salesman construction, which is of independent interest.

It is known that everyGδsubsetEof the plane containing a dense set of lines, even if it has measure zero, has the property that every real-valued Lipschitz function onℝ2has a point of differentiability inE. Here we show that the set of points of differentiability of Lipschitz functions inside such sets may be surprisingly tiny: we construct aGδsetE⊂ℝ2containing a dense set of lines for which there is a pair of real-valued Lipschitz functions onℝ2having no common point of differentiability inE, and there is a real-valued Lipschitz function onℝ2whose set of points of differentiability inEis uniformly purely unrectifiable.

We study inverse potential problems with source term the divergence of some unknown (ℝ3-valued) measure supported in a plane; e.g., inverse magnetization problems for thin plates. We investigate methods for recovering a magnetization μ by penalizing the measure-theoretic total variation norm ∥μ∥TV , and appealing to the decomposition of divergence-free measures in the plane as superpositions of unit tangent vector fields on rectifiable Jordan curves. In particular, we prove for magnetizations supported in a plane that TV -regularization schemes always have a unique minimizer, even in the presence of noise. It is further shown that TV -norm minimization (among magnetizations generating the same field) uniquely recovers planar magnetizations in the following two cases: (i) when the magnetization is carried by a collection of sufficiently separated line segments and a set that is purely 1-unrectifiable; (ii) when a superset of the support is tree-like. We note that such magnetizations can be recovered via TV -regularization schemes in the zero noise limit by taking the regularization parameter to zero. This suggests definitions of sparsity in the present infinite dimensional context, that generate results akin to compressed sensing.

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 locally describes their directions – defined by a universal property in an appropriate category that we introduce. These methods allow to promote each measure space (X, A , µ) to a strictly localizable version (X̂, Â, µ̂), so that the dual of L1 (X, A , µ) is L∞ (X̂, Â, µ̂). 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.

Abstract We obtain a criterion for an analytic subset of a Euclidean space to contain points of differentiability of a typical Lipschitz function: namely, that it cannot be covered by countably many sets, each of which is closed and purely unrectifiable (has a zero-length intersection with every $C^1$ curve). Surprisingly, we establish that any set failing this criterion witnesses the opposite extreme of typical behaviour: in any such coverable set, a typical Lipschitz function is everywhere severely non-differentiable.

Cette thèse étudie différentes variantes de la mesure harmonique et leurs relations avec la géométrie de la frontière d'un domaine. Dans la première partie de la thèse, on se concentre sur l'analogue de la mesure harmonique pour les domaines ayant des frontières de dimensions plus petites, définies via la théorie des opérateurs elliptiques dégénérés récemment développée par David et al. Plus précisément, on démontre qu'il n'existe pas de famille à un paramètre non dégénérée de solutions de l'équation LμDμ = 0, ce qui constitue la première étape pour retrouver une forme de l'assertion "si la fonction de distance à la frontière d'un domaine est harmonique, alors la frontière est plate", qui manque à la théorie des opérateurs elliptiques dégénérés. On découvre et explique également pourquoi la stratégie la plus naturelle pour étendre notre résultat à l'absence de solutions individuelles de l'équation LμDμ = 0 ne fonctionne pas. Dans la deuxième partie de la thèse, on s'intéresse aux mesures elliptiques dans le cadre classique. On construit une nouvelle famille d'opérateurs avec des coefficients continus scalaires dont les mesures elliptiques sont absolument continues par rapport aux mesures de Hausdorff sur des flocons de neige symétriques de type Koch. Cette famille enrichit la collection des exemples connus de mesures elliptiques qui se comportent très différemment de la mesure harmonique et des mesures elliptiques d'opérateurs proches, d'une certaine manière, du Laplacien. De plus, nos nouveaux exemples ne sont pas compacts. La construction fournit également une méthode possible pour construire des opérateurs ayant ce type de comportement pour d'autres fractales qui possèdent suffisamment de symétries
This thesis studies different counterparts of the harmonic measure and their relations with the geometry of the boundary of a domain. In the first part of the thesis, we focus on the analogue of harmonic measure for domains with boundaries of smaller dimensions, defined via the theory of degenerate elliptic operators developed recently by David et al. More precisely, we prove that there is no non-degenerate one-parameter family of solutions to the equation LμDμ = 0, which constitutes the first step to recover an analogue of the statement ``if the distance function to the boundary of a domain is harmonic, then the boundary is flat'', missing from the theory of degenerate elliptic operators. We also find out and explain why the most natural strategy to extend our result to the absence of individual solutions to the equation LμDμ = 0 does not work. In the second part of the thesis, we focus on elliptic measures in the classical setting. We construct a new family of operators with scalar continuous coefficients whose elliptic measures are absolutely continuous with respect to the Hausdorff measures on Koch-type symmetric snowflakes. This family enriches the collection of a few known examples of elliptic measures which behave very differently from the harmonic measure and the elliptic measures of operators close in some sense to the Laplacian. Plus, our new examples are non-compact. Our construction also provides a possible method to construct operators with this type of behaviour for other fractals that possess enough symmetries