Contents Preface page xiii Notation xvii PART I RADON MEASURES ON R n 1 1 Outer measures 4 1.1 Examples of outer measures 4 1.2 Measurable sets and σ-additivity 7 1.3 Measure Theory and integration 9 2 Borel and Radon measures 14 2.1 Borel measures and Carathéodory's criterion 14 2.2 Borel regular measures 16 2.3 Approximation theorems for Borel measures 17 2.4 Radon measures. Restriction, supp...

We show that if n ≥ 1, Ω ⊂ R is a connected domain that is porous around a subset E ⊂ ∂Ω of finite and positive Hausdorff H-measure, and the harmonic measure ω is absolutely continuous with respect to H on E, then ω|E is concentrated on an n-rectifiable set.

We investigate the asymptotic behavior, as N grows, of the largest minimal pairwise distance of N points restricted to an arbitrary compact rectifiable set embedded in Euclidean space, and we find the limit distribution of such optimal configurations. For this purpose, we compare best-packing configurations with minimal Riesz s-energy configurations and determine the s-th root asymptotic behavi...

The space SBV of special BV functions whose gradient measure has no Cantor part was singled out by De Giorgi and Ambrosio [4] as the natural setting to study variational problems where both volume and surface densities have to be taken into account. In fact, for a SBV function f the derivative Df is the sum of a measure Daf absolutely continuous with respect to the Lebesgue measure and a singul...

We prove that parabolic uniformly rectifiable sets admit (bilateral) corona decompositions with respect to regular Lip(1,1/2) graphs. Together our previous work, this allows us conclude if $$\Sigma \subset {\mathbb {R}}^{n+1}$$ is Ahlfors-David regular, then the following statements are equivalent.

Given a compact d-rectifiable set A embedded in Euclidean space and a distribution ρ(x) with respect to d-dimensional Hausdorff measure on A, we address the following question: how can one generate optimal configurations of N points on A that are “well-separated” and have asymptotic distribution ρ(x) as N → ∞? For this purpose we investigate minimal weighted Riesz energy points, that is, points...

We prove that if u is the entropy solution to a scalar conservation law in one space dimension, then the entropy dissipation is a measure concentrated on countably many Lipschitz curves. This result is a consequence of a detailed analysis of the structure of the characteristics. In particular the characteristic curves are segments outside a countably 1-rectifiable set and the left and right tra...

The distributional k-dimensional Jacobian of a map u in the Sobolev space W 1,k−1 which takes values in the the sphere S can be viewed as the boundary of a rectifiable current of codimension k carried by (part of) the singularity of u which is topologically relevant. The main purpose of this paper is to investigate the range of the Jacobian operator; in particular, we show that any boundary M o...