The first attempt corresponds to the situation in which one seeks the lowest point in a very irregular mountain landscape. One has always to take long detours around obstacles and, being situated in a deep valley, it is very difficult to recognize if the ones behind the mountains are even deeper. In the second approach one first averages over the rapid oscillation between highs and lows, and obtains a much smoother and clearer landscape, in which the lowest point can easily be found. Such averaging or coarse-graining procedures have been intensely studied in theoretical physics and in many applications. It is however very important to consider in any particular case the specific properties of the considered system, e.g. of the shape-memory alloys.
Young measures are also helpful for a better understanding of microstructure in crystals.
However, in this case there is an additional difficulty, in that two different crystalline structures can be joined together only if the two lattices agree on the interface. This geometric condition can be abstractly included in the theory in a simple way, but many relevant questions are still open. This apparently harmless geometric condition leads naive averaging methods to failure, because in the averaging geometrically prohibited structures can arise. In the last years it has become clear that this problem is strictly connected with fundamental problems from other areas of mathematics. One question which can be easily understood is the following.