Geometric rigidity for incompatible fields in the multi-well case and an application to strain-gradient plasticity

We derive a quantitative rigidity estimate for a multiwell problem in nonlinear elasticity with dislocations. Precisely, we show that the L^{1^{*}}-distance of a possibly incompatible strain field from a single well is controlled in terms of the L^{1^{*}}-distance from a finite set of wells, of curl \beta and of div \beta. As a consequence, we derive a strain-gradient plasticity model as Gamma-limit of a nonlinear finite dislocation model, containing a singular perturbation term accounting for the divergence of the strain field. This can also be seen as a generalization of the result of Alicandro et al. (2018) to the case of incompatible vector fields.