MiS Preprint Repository

We study the interaction of a singularly perturbed multiwell energy (with an anisotropic nonlocal regularizing term of $H^{1/2}$ type) and a pinning condition. This functional arises in a phase field model for dislocations which was recently proposed by Koslowski, Cuitiño and Ortiz but is also of broader mathematical interest. In the context of the dislocation model we identify the $\mathrm{\Gamma }$-limit of the energy in all scaling regimes for the number $N_{\epsilon }$ of obstacles. The most interesting regime is $N_{\epsilon }\approx |\ln \epsilon |/\epsilon$, where $\epsilon$ is a nondimensional length scale related to the size of the crystal lattice. In this case the limiting model is of line tension type. One important feature of our model is that the set of energy wells is periodic and hence not compact. A key ingredient in the proof is thus a compactness estimate (up to a single translation) for finite energy sequences, which generalizes earlier results of Alberti, Bouchitté and Seppecher for the two-well problem with an $H^{1/2}$ regularization.