We continue to look on the problem
Are two optimal holes better than one?
Again, the answer depends on the sign of a. Now we consider:
Loadings of different signs
In this case, the two optimal holes are better than one optimal hole!
Both optimal holes in the pair remain elongated quadrangles. The functional is sensitive to both shapes and mutual positions of the two holes.
Continuing, we conclude that three holes perform better than two, and so on. This time, the optimality of an assembly of holes determines their shapes and distances between them.
This effect demonstrates that the compexity of a simple system can grow due to an optimality requirement.Interestingly, the system of several optimal inclusions forms a complicated cluster of shapes while the loading is homogeneous and the plane is infinite!
The very best configuration corresponds to infinitely many inclusions.
The optimal configuration of the finite number of holes and their shapes is more sophisticated than the limiting cases of one optimal "square" hole and of infinite cluster of them. Yury Grabovsky uses the following joke about mathematicians to illustrate the difficulty of the intermediate cases:
A mathematician is asked to design a table. He first designs a table with no legs. Then he designs a table with infinitely many legs. He spend the rest of his life generalizing the results for the table with N legs (where N is not necessarily a natural number). Look here for similar stories.