Calculus of Variations and Geometric Measure Theory

S. Conti - A. Garroni - A. Massaccesi

Modeling of dislocations and relaxation of functionals on 1-currents with discrete multiplicity

created by massaccesi on 29 Nov 2013
modified by garroni on 18 Sep 2020


Published Paper

Inserted: 29 nov 2013
Last Updated: 18 sep 2020

Journal: Calc. Var. Partial Differential Equations
Volume: 54
Number: 2
Pages: 1847-1874
Year: 2015

ArXiv: 1409.6084 PDF


In the modeling of dislocations one is lead naturally to energies concentrated on lines, where the integrand depends on the orientation and on the Burgers vector of the dislocation, which belongs to a discrete lattice. The dislocations may be identified with divergence-free matrix-valued measures supported on curves or with 1-currents with multiplicity in a lattice. In this paper we develop the theory of relaxation for these energies and provide one physically motivated example in which the relaxation for some Burgers vectors is nontrivial and can be determined explicitly. From a technical viewpoint the key ingredients are an approximation and a structure theorem for 1-currents with multiplicity in a lattice.