Coupled singular perturbations and homogenization in phase transitions

This thesis addresses the characterization of the limiting energy of families of functionals of the type J3u :=13E3 u with Eepsilon of the form E3u :=Wf3&parl0; x,ux,31u x&parr0;dx.;Motivated by issues in fluid-fluid phase transition problems, and the corresponding development of mathematical techniques to treat (anisotropic) singular perturbations, here the focus will be on special classes of non-negative energy densities fepsilon satisfying fepsilon(x, u, 0) = 0 if and only if u ∈ {a, b}.;The first part of this work addresses the case where the singular perturbation is coupled with the two-well energy, and the second part considers (decoupled) periodically oscillating two-well energy density and second order penalization terms.