In this paper we examine nonlinear integrodifferential inclusions defined in a separable Banach space. Using a compactness type hypothesis involving the ball measure of noncompactness, we establish two existence results. One involving convex-valued orientor fields and the other nonconvex valued ones.

Let Γ ( E ) $\Gamma (E)$ be the family of all paths which meet a set in metric measure space X. The function ↦ A M $E \mapsto AM(\Gamma (E))$ defines $AM$ -modulus X where refers to approximation modulus [22]. We compare $AM(\Gamma Hausdorff c o H 1 $co\mathcal {H}^1(E)$ codimension one and show that ≈ \begin{equation*}\hskip6pc co\mathcal {H}^1(E) \approx (E))\hskip-6pc \end{equation*} for Sus...

Let d ≥ 1 be an integer and E a self-similar fractal set, which is the attractor of a uniform contracting iterated function system (UIFS) on Rd. Denote by D the Hausdorff dimension, by HD(E) the Hausdorff measure and by diam(E) the diameter of E. If the UIFS is parametrised by its contracting factor c, while the set ω of fixed points of the UIFS does not depend on c, we will show the existence ...

In this paper, we consider generic corank 2 sub-Riemannian structures, and we show that the Spherical Hausdorf measure is always a C-smooth volume, which is in fact generically Csmooth out of a stratified subset of codimension 7. In particular, for rank 4, it is generically C 2 . This is the continuation of a previous work by the auhors. subjclass: 53C17, 49J15, 58C35

Let E be a set in R with finite n-dimensional Hausdorff measure H such that lim infr→0 r −n H (B(x, r)∩E) > 0 for H-a.e. x ∈ E. In this paper it is shown that E is n-rectifiable if and only if

(i= 1 . 2, . . .) such that the intersection n A, contains a perfect subset i=1 (and is therefore of power 2No) . They asked for what Hausdorff measure functions (k(i) is it possible to choose the subsequence to make the intersection set (1 A„,, of positive -measure . In the present note We show that the strongest possible result in this direction is true . This is given by the following; theor...

In the present paper we prove that for any open connected set Ω ⊂ R, n ≥ 1, and any E ⊂ ∂Ω with 0 < H(E) < ∞ absolute continuity of the harmonic measure ω with respect to the Hausdorff measure on E implies that ω|E is rectifiable. CONTENTS

In this paper we study the nilpotent 2-step, corank 2 sub-Riemannian metrics that are nilpotent approximations of general sub-Riemannian metrics. We exhibit optimal syntheses for these problems. It turns out that in general the cut time is not equal to the first conjugate time but has a simple explicit expression. As a byproduct of this study we get some smoothness properties of the spherical H...

Recently, the theory of quasiregular mappings on Carnot groups has been developed intensively. Let ν stand for the homogeneous dimension of a Carnot group and let m be the index of the last vector space of the corresponding Lie algebra. We prove that the (ν −m− 1)-dimensional Hausdorff measure of the image of the branch set of a quasiregular mapping on the Carnot group is positive. Some estimat...