A Variational Approach to Single Crystals with Dislocations

We study the graphs of maps $u:\Omega\rightarrow\Bbb R^3$ whose curl is an integral 1-current with coefficients in $\Bbb Z^3$. We characterize the graph boundary of such maps under a suitable summability property. We apply these results to study a three-dimensional single crystal with dislocations forming general one-dimensional clusters in the framework of finite elasticity. By virtue of a variational approach, a free energy depending on the deformation field and its gradient is considered. The problem we address is the joint minimization of the free energy with respect to the deformation field and the dislocation lines. We apply closedness results for graphs of torus-valued maps, seen as integral currents and, from the characterization of their graph boundaries, we are able to prove existence of minimizers.