On the lower semicontinuous envelope of functionals defined on polyhedral chains

@article{Colombo2017OnTL,
  title={On the lower semicontinuous envelope of functionals defined on polyhedral chains},
  author={Maria Colombo and Antonio De Rosa and Andrea Marchese and Salvatore Stuvard},
  journal={Nonlinear Analysis-theory Methods \& Applications},
  year={2017},
  volume={163},
  pages={201-215}
}
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, subadditive, and lower semicontinuous function with H ( 0 ) = 0 , and by Φ H the functional induced by H on polyhedral m -chains, namely Φ H ( P ) ≔ ∑ i = 1 N H ( θ i ) H m ( σ i ) , for every  P = ∑ i = 1 N θ i 〚 σ i 〛 ∈ P m ( R n ) , we prove that the lower semicontinuous envelope of Φ H coincides… 
Strong approximation in h-mass of rectifiable currents under homological constraint
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 |
Variational approximation of size-mass energies for k-dimensional currents
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
A Multimaterial Transport Problem and its Convex Relaxation via Rectifiable G-currents
TLDR
This paper proposes an Eulerian formulation of the discrete problem, describing the flow of each commodity through every point of the network, and proves that the problem can be rephrased as a mass minimization problem in a class of rectifiable currents with coefficients in a group, allowing to introduce a notion of calibration.
General transport problems with branched minimizers as functionals of 1-currents with prescribed boundary
A prominent model for transportation networks is branched transport, which seeks the optimal transportation scheme to move material from a given initial to a given final distribution. The cost of the
A phase-field approximation of the Steiner problem in dimension two
Abstract In this paper we consider the branched transportation problem in two dimensions associated with a cost per unit length of the form 1 + β ⁢ θ {1+\beta\,\theta} , where θ denotes the amount of
The oriented mailing problem and its convex relaxation
In this note we introduce a new model for the mailing problem in branched transportation in order to allow the cost functional to take into account the orientation of the moving particles. This gives
Approximation of rectifiable 1-currents and weak-⁎ relaxation of the h-mass
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
On the Well‐Posedness of Branched Transportation
TLDR
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.
Improved stability of optimal traffic paths
Models involving branched structures are employed to describe several supply-demand systems such as the structure of the nerves of a leaf, the system of roots of a tree, and the nervous or
Stability for the mailing problem
We prove that optimal traffic plans for the mailing problem in $\mathbb{R}^d$ are stable with respect to variations of the given coupling, above the critical exponent $\alpha=1-1/d$, thus solving an
...
1
2
...

References

SHOWING 1-10 OF 17 REFERENCES
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
Size minimization and approximating problems
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
A simple phase-field approximation of the Steiner problem in dimension two
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
Rectifiability of flat chains
We prove (without using Federer's structure theorem) that a finite-mass flat chain over any coefficient group is rectifiable if and only if almost all of its 0-dimensional slices are rectifiable.
General transport problems with branched minimizers as functionals of 1-currents with prescribed boundary
A prominent model for transportation networks is branched transport, which seeks the optimal transportation scheme to move material from a given initial to a given final distribution. The cost of the
On the Lagrangian branched transport model and the equivalence with its Eulerian formulation
First we present two classical models of Branched Transport: the Lagrangian model introduced by Bernot, Caselles, Morel, Maddalena, Solimini, and the Eulerian model introduced by Xia. An emphasis is
Geometric Integration Theory
Basics.- Caratheodory's Construction and Lower-Dimensional Measures.- Invariant Measures and the Construction of Haar Measure..- Covering Theorems and the Differentiation of Integrals.- Analytical
OPTIMAL PATHS RELATED TO TRANSPORT PROBLEMS
In transport problems of Monge's types, the total cost of a transport map is usually an integral of some function of the distance, such as |x - y|p. In many real applications, the actual cost may
Improved stability of optimal traffic paths
Models involving branched structures are employed to describe several supply-demand systems such as the structure of the nerves of a leaf, the system of roots of a tree, and the nervous or
Geometric Measure Theory
Introduction Chapter 1 Grassmann algebra 1.1 Tensor products 1.2 Graded algebras 1.3 Teh exterior algebra of a vectorspace 1.4 Alternating forms and duality 1.5 Interior multiplications 1.6 Simple
...
1
2
...