Problemi di interazione non-locale in Teoria delle Dislocazioni

In the thesis we study variational problems arising from Dislocation Theory. In the first part we prove that a proper scaling of the energy I_n of n dislocations in R^2 Gamma-converges to a functional defined over the space of probability measures of R^2. Moreover, we show that the sequence of Gibbs measures induced by (I_n) satisfies a Large Deviation Principle. In the second and third chapter we study two variational problems corresponding to the Gamma-limit. For some particular cases, we manage to find explicitly the minimizer. In the last chapter we describe a numerical method for the approximation of minimizers when explicit computations seem to be difficult to carry out.