Abstract We study some fundamental properties of real rectifiable currents and give a generalization King’s theorem to characterize defined by positive holomorphic chains. Our main tool is Siu’s semi-continuity our proof largely simplifies proof. A consequence this result sufficient condition for the Hodge conjecture.

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in unit disk D is due to dissertation Luzin. paper [11] was devoted continuous boundary quasilinear Poisson equations smooth (C1) domains. present (over natural parameter) any Jordan domains rectifiable boundaries. For this purpose, it constructed completely operators generating nonclassical solutions bound...

In this note we show that for every measurable function on $\mathbb{R}^n$ the set of points where blowup exists and is not constant $(n-1)$-rectifiable. particular, $u\in L^1_{loc}(\mathbb{R}^n)$ jump $J_u$

The Mα energy which is usually minimized in branched transport problems among singular 1-dimensional rectifiable vector measures with prescribed divergence is approximated (and convergence is proved) by means of a sequence of elliptic energies, defined on more regular vector fields. The procedure recalls the Modica-Mortola one for approximating the perimeter, and the double-well potential is re...

In the present paper we sketch the proof of the fact that for any open connected set Ω ⊂ Rn+1, 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.