Analytic and geometric properties of dislocation singularities

Abstract

AbstractThis paper deals with the analysis of the singularities arising from the solutions of the problem${-}\,{\rm Curl\ } F=\mu $, whereFis a 3 × 3 matrix-valuedLp-function ($1\les p<2$) andμa 3 × 3 matrix-valued Radon measure concentrated in a closed loop in Ω ⊂ ℝ3, or in a network of such loops (as, for instance, dislocation clusters as observed in single crystals). In particular, we study the topological nature of such dislocation singularities. It is shown that$F=\nabla u$, the absolutely continuous part of the distributional gradientDuof a vector-valued functionuof special bounded variation. Furthermore,ucan also be seen as a multi-valued field, that is, can be redefined with values in the three-dimensional flat torus 𝕋3and hence is Sobolev-regular away from the singular loops. We then analyse the graphs of such maps represented as currents in Ω × 𝕋3and show that their boundaries can be written in term of the measureμ. 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. In the spirit of previous work, we finally give some examples of variational problems where such results provide existence of solutions.