Small Oscillation Coverings and the Excision Method

Abstract

In this chapter, we develop the main general tool used for proving regularity properties of the minimizers of the Mumford-Shah functional. It will be called Excision Method and consists in showing that, whenever the set K is too “ragged” inside some disk D, and whenever u is smooth enough around each piece of K, then the excision operation, consisting of removing from the edge set K the points contained in D, decreases the value of the functional E(K).

This kind of information will have a straightforward use: since the functional E(K) cannot decrease when the edge set K is a minimum, we shall conclude that K is not too “ragged”, whatever the disk D is, and regularity properties for K will follow. Let us now see roughly which kind of raggedness condition we shall consider. We shall assume that the edge set can be covered by a family of sets D_i on whose boundary, ∂D_i, the function u = u k does not oscillate much. Such coverings will be called small oscillation coverings. We shall prove that when one can cover the edge set K with a small oscillation covering made of sets D_i whose size is small enough with respect to the diameter of D, then the excision operation works successfully and decreases the Mumford-Shah energy.

Several applications of this general excision principle will be done in the next two chapters. The “concrete properties” of the minimizers which will be proved therein will in fact be straightforward applications of the excision method.