On explicit solutions for the problem of Mumford and Shah

We look for explicit image segmentations in the framework of the variationa l model proposed by Mumford and Shah. We first treat the symmetric case whe n the "screen" is a disk Omega and the image is a concentric disk D subset of Omega. We prove the optimal segmentation is either the given disk D or t he solution of the associated Neumann problem, depending on both the differ ence of intensity between the background and the disk, and the distance sep arating partial derivative Omega and partial derivative D. Both segmentatio ns are optimal in some critical cases which we characterize. Our main resul t is a first step towards a generalization of this behaviour. In case Omega and D are convex, we prove the following for an optimal segmentation (u, K ) such that K subset of D: K tends to partial derivative D (in the Hausdorf f distance) when the difference of intensity between Omega and D goes to in finity.