We consider the class of integer rectifiable currents without boundary in Rn × R satisfying a positivity condition. We establish that these currents can be written as a linear superposition of graphs of finitely many functions with bounded variation.

Abstract We extend Korevaar–Schoen’s theory of metric valued Sobolev maps to cover the case source space being an $$\mathsf{RCD}$$ RCD space. In this situation it appears that no version ‘subpartition lemma’ holds: obtain both existence limit approximated energies and lower semicontinuity energy we shall rely...

We consider a class of variational problems for differential inclusions, related to the control of wild fires. The area burned by the fire at time t > 0 is modelled as the reachable set for a differential inclusion ẋ ∈ F (x), starting from an initial set R0. To block the fire, a barrier can be constructed progressively in time. For each t > 0, the portion of the wall constructed within time t i...

Numerous elliptic and parabolic variational problems arising in physics and geometry (Ginzburg-Landau equations, harmonic maps, Yang-Mills fields, Omega-instantons, Yamabe equations, geometric flows in general...) possess a critical dimension in which an invariance group (similitudes, conformal groups) acts. This common feature generates, in all these different situations, the same non-linear e...

In this paper we focus on characterizing compact subsets of curves in R with regular derivatives. Namely, we give characterization of compact subsets of finite sums of disjoint finite-length curves with ω-continuous derivative and without selfintersections. Intuitively, our condition can be formulated as there exists a finite set of regular curves covering a compact set K iff every triple of po...

Let G be a connected semisimple Lie group such that the associated symmetric space X is Hermitian and let Γg be the fundamental group of a compact orientable surface of genus g ≥ 2. We survey the study of maximal representations of Γg into G that is the subset of Hom(Γg, G) which is a union of components characterized by the maximality of the Toledo invariant ([16] and [14]). Then we concentrat...