We establish a framework to construct a global solution in the space of finite energy to a general form of the Landau-Lifshitz-Gilbert equation in R2. Our characterization yields a partially regular solution, smooth away from a 2-dimensional locally finite Hausdorff measure set. This construction relies on approximation by discretization, using the special geometry to express an equivalent syst...

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