Variational Methods for Second Order Structured Deformations and Multiscale Problems

In this thesis, a family of integral representation results are proved for problems involving energies with different dimensionalities and multiscale interactions. The first part is work in the framework of functions of bounded Hessian, with an eye towards application to the theory of second order structured deformations. A relaxation theorem for BH functionals is obtained in the spirit of the 1992 work of Ambrosio and Dal Maso, Fonseca and Müller within the BV context. An integral representation theorem is established for abstract second order structured deformations functionals, using proof techniques from the global method for integral representation introduced in 1998 by Bouchitté, Fonseca and Mascarenhas. The second family of results concerns systems featuring simultaneous homogenization and phase transition effects. These are studied via the technique of 􀀀-convergence, and multiple regimes
are considered corresponding to the relative scaling of the phase transition thickness and the scale of the heterogeneity. In particular, 􀀀-limit results are proved in the general case of vector-valued functions when the two rates are commensurate and when the frequency of the
heterogeneity is sufficiently small with respect to the thickness.