# 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}, journal={Advances in Calculus of Variations}, year={2019}, volume={14}, pages={343 - 363} }

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}_…

## 7 Citations

Approximation of rectifiable 1-currents and weak-⁎ relaxation of the h-mass

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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

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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, MathematicsESAIM: Control, Optimisation and Calculus of Variations
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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

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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…

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- Computer Science, MathematicsArXiv
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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
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of Branched Transportation

- Mathematics, Computer ScienceCommunications on Pure and Applied Mathematics
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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.

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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.…

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