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This part mainly deals with the existence (and uniqueness, if possible) of optimal shapes and their regularity. The topics have been marked by the pioneering work of Caffarelli and Alt, and presently are intensively developed, in particular by French and Italian schools. In general, the problem of the existence of an optimal shape is complex and difficult due to the lack of appropriate compactness in the set of domains and to poor behavior of the shape functionals with respect to the topology in the set of domains While the uniqueness of optimal shape does not hold in general, there are fine regularity results of optimal shapes, see the works of L.A. Caffarelli, mostly based on a geometrical approach, and for more recent results based on an analytical approach the reader may consider and the references within.
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