# 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…

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## References

SHOWING 1-10 OF 17 REFERENCES

The deformation theorem for flat chains

- Mathematics
- 1999

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

- 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…

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…

Rectifiability of flat chains

- Mathematics
- 1999

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

- Mathematics
- 2017

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

- Mathematics
- 2015

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

- Mathematics
- 2008

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

- Mathematics
- 2003

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

- Mathematics
- 2017

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

- Mathematics
- 1969

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…