Analytic and geometric properties of dislocation singularities

This paper deals with the analysis of the singularities arising from the solutions of the problem Curl F = µ, where F is a 3 × 3 matrix-valued L p-function (1 ≤ p < 2) and µ a 3 × 3 matrix-valued Radon measure concentrated in a closed loop in Ω ⊂ R 3 , or in a network of such loops, i.e., similar to dislocation clusters as observed in single crystals. In particular, we study the topological nature of such dislocation singularities. It is shown that F = ∇u, the absolutely continuous part of the distributional gradient Du of a vector-valued function u of special bounded variation. Furthermore, u can be redefined with values in the three-dimensional torus T 3 and hence is Sobolev-regular away from the singular loops. We then analyze the graphs of such maps represented as currents on Ω × T 3 and show that their boundaries can be written in term of the measure µ. Readapting some well-known results for Cartesian currents, we recover closure and compactness properties of the class of maps with bounded curl concentrated on dislocation networks. We finally give some examples of variational problems where such results provide existence of solutions.