Strong approximation in h-mass of rectifiable currents under homological constraint

@article{Chambolle2019StrongAI,
title={Strong approximation in h-mass of rectifiable currents under homological constraint},
author={A. Chambolle and Luca Alberto Davide Ferrari and Beno{\^i}t Merlet},
year={2019},
volume={14},
pages={343 - 363}
}
• Published 13 June 2018
• Physics, Mathematics
• Advances in Calculus of Variations
Abstract Let h : ℝ → ℝ + {h:\mathbb{R}\to\mathbb{R}_{+}} be a lower semicontinuous subbadditive and even function such that h ⁢ ( 0 ) = 0 {h(0)=0} and h ⁢ ( θ ) ≥ α ⁢ | θ | {h(\theta)\geq\alpha|\theta|} for some α > 0 {\alpha>0} . If T = τ ⁢ ( M , θ , ξ ) {T=\tau(M,\theta,\xi)} is a k-rectifiable chain, its h-mass is defined as 𝕄 h ⁢ ( T ) := ∫ M h ⁢ ( θ ) ⁢ 𝑑 ℋ k . \mathbb{M}_{h}(T):=\int_{M}h(\theta)\,d\mathcal{H}^{k}. Given such a rectifiable flat chain T with 𝕄 h ⁢ ( T ) < ∞ {\mathbb{M}_…

Figures from this paper

Approximation of rectifiable 1-currents and weak-⁎ relaxation of the h-mass
• Mathematics
Journal of Mathematical Analysis and Applications
• 2019
Based on Smirnov's decomposition theorem we prove that every rectifiable $1$-current $T$ with finite mass $\mathbb{M}(T)$ and finite mass $\mathbb{M}(\partial T)$ of its boundary $\partial T$ can be
A multi-material transport problem with arbitrary marginals
• Mathematics
• 2018
In this paper we study general transportation problems in $\mathbb{R}^n$, in which $m$ different goods are moved simultaneously. The initial and final positions of the goods are represented by
Variational approximation of size-mass energies for k-dimensional currents
• Physics, Mathematics
ESAIM: Control, Optimisation and Calculus of Variations
• 2019
In this paper we produce a Γ-convergence result for a class of energies Fε,ak modeled on the Ambrosio-Tortorelli functional. For the choice k = 1 we show that Fε,a1 Γ-converges to a branched
Recent results on non-convex functionals penalizing oblique oscillations
• Physics
• 2019
The aim of this note is to review some recent results on a family of functionals penalizing oblique oscillations. These functionals naturally appeared in some varia-tional problem related to pattern
An adaptive finite element approach for lifted branched transport problems
• Computer Science, Mathematics
ArXiv
• 2020
An efficient numerical treatment based on a specifically designed class of adaptive finite elements allows the computation of finely resolved optimal transportation networks despite the high dimensionality of the convex optimization problem and its complicated set of nonlocal constraints.
On the Well‐Posedness of Branched Transportation
• Mathematics, Computer Science
Communications on Pure and Applied Mathematics
• 2020
It is proved that any limit of optimal traffic paths is optimal as well, which solves an open problem in the field and shows in full generality the stability of optimal transport paths in branched transport.
Space-time integral currents of bounded variation
Motivated by a recent model for elasto-plastic evolutions that are driven by the flow of dislocations [12], this work develops a theory of space-time integral currents with bounded variation in time.

References

SHOWING 1-10 OF 38 REFERENCES
Connecting measures by means of branched transportation networks at finite cost
• Mathematics
• 2009
We study the couples finite Borel measures φ0 and φ1 with compact support in $$\mathbb{R}^n$$ which can be transported to each other at a finite Wα cost, where  W^{\alpha } \left( {\varphi_0,
On the lower semicontinuous envelope of functionals defined on polyhedral chains
• Mathematics
• 2017
Abstract In this note we prove an explicit formula for the lower semicontinuous envelope of some functionals defined on real polyhedral chains. More precisely, denoting by H : R → [ 0 , ∞ ) an even,
Uniform estimates for a Modica-Mortola type approximation of branched transportation
Models for branched networks are often expressed as the minimization of an energy $M^\alpha$ over vector measures concentrated on $1$-dimensional rectifiable sets with a divergence constraint. We
The deformation theorem for flat chains
We prove that the deformation procedure of Federer and Fleming gives good approximations to arbritrary flat chains, not just those of finite mass and boundary mass. This implies, for arbitrary
Variational approximation of size-mass energies for k-dimensional currents
• Physics, Mathematics
ESAIM: Control, Optimisation and Calculus of Variations
• 2019
In this paper we produce a Γ-convergence result for a class of energies Fε,ak modeled on the Ambrosio-Tortorelli functional. For the choice k = 1 we show that Fε,a1 Γ-converges to a branched
Normal Currents: Structure, Duality Pairings and div–curl Lemmas
Abstract.The paper gives a decomposition of a general normal r-dimensional current [5] into the sum of three measures of which the first is an r-dimensional rectifiable measure, the second is the
Size minimization and approximating problems
• Mathematics
• 2003
Abstract.We consider Plateau type variational problems related to the size minimization of rectifiable currents. We realize the limit of a size minimizing sequence as a stationary varifold and a
Approximation of Length Minimization Problems Among Compact Connected Sets
• Mathematics, Computer Science
SIAM J. Math. Anal.
• 2015
This paper provides an approximation of some classical minimization problems involving the length of an unknown one-dimensional set, with an additional connectedness constraint, in dimension two, and introduces a term of new type relying on a weighted geodesic distance that forces the minimizers to be connected at the limit.
A simple phase-field approximation of the Steiner problem in dimension two
• Mathematics
• 2016
In this paper we consider the branched transportation problem in 2D associated with a cost per unit length of the form 1 + αm where m denotes the amount of transported mass and α > 0 is a fixed
Remplissage De L'Espace Euclidien Par Des Complexes PolyÉdriques D'Orientation ImposÉe Et De RotonditÉ Uniforme
We build polyhedral complexes in R that coincide with dyadic grids with different orientations, while keeping uniform lower bounds (depending only on n) on the flatness of the added polyhedrons