Phase Transitions and Microstructure
Pablo Pedregal · Progress in nonlinear differential equations and their applications · 1997
We have tried to emphasize in the previous chapter the importance of the study of variational principles for which some lack of convexity leads one to consider the behavior of minimizing sequences. From the mathematical point of view, there are two ways to proceed whenever there are no minimizers as a consequence of this lack of convexity. One is to “convexify” the energy density itself or the nonconvex constraints involved in order to obtain a new functional which can be analyzed through the techniques dicussed in Chapter 3. The task is to relate the information concerning this convexified functional with the original one. Relaxation theorems refer to this issue. Another possibility is to enlarge sufficiently the class of competing objects in some kind of generalized variational setting as to include minimizers. These generalized objects are parametrized measures. They were introduced by Young in this same context to understand ill-posed variational problems. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.