The appearance of microstructures in problems with incompatible wells and their numerical approach

The goal of the paper is to analyze the creation of microstructure in problems of Calculus of Variations with wells. More precisely we consider a case with strong incompatibility between the wells. This forces the minimizing sequences to use other gradients than the wells in a puzzling way. Using a $P_1$ method we are then able to single out discrete minimizing sequences and to give energy estimates in terms of the mesh size.