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We show how it is possible to approximate the Mumford-Shah (see [29]) image segmentation functional uE W 1 . * ( Q \ K ) , K C Q closedin Q by elliptic functionals defined on Sobolev spaces. The heuristic idea is to consider functionals Sh( u, z ) with z ranging between 0 and I and related to the set K. The minimizing zh are near to 1 in a neighborhood of the set K, and far from the neighborhood they are very small. The neighborhood shrinks as h + +co .For a similar approach to the problem compare Kulkarni; see [25]. The approximation of Sh to 9 takes place in a variational sense, the De Giorgi I'-convergence. L. AMBROSIO AND V. M. TORTORELLIwhere a, / 3 > 0 are fixed parameters, and g E L*(fl). In the case n = 2 this functional has been suggested by Mumford-Shah (see [29]) for a variational a p proach to image segmentation. The function g in (2) represents the image of a group of objects given by a camera, with discontinuities along the edges of the objects. By minimizing the functional (2) one tries to distinguisl? the discontinuities due to the edges and shadows from the discontinuities due to noise and small irregularities. The functional penalizes large sets K, and outside the set K the function u is required to be close to g and W1.2. In the papers [29], [ 301 Mumford-Shah conjecture that 9 has minimizers, and they study their behavior under some a priori regularity assumptions. Recently, the Mumford-Shah conjecture concerning existence of minimizers has been proved for general integers n by De Giorgi-Camiero-Leaci; see [ 17 1. In the case n = 2 a different proof has been discovered by Dal Maso-Morel-Solimini; see [ 131. The idea in common to both proofs is to use a weak formulation of the problem by setting where S, is the discontinuity set of u in an approximate sense, and u varies in a special class of functions of bounded variation, denoted by S B V ( Q ) . This class consists of all functions of bounded variation such that the distributional derivative is absolutely continuous with respect to Lebesgue measure plus an ( n -1 )-dimensional measure. Since (see [2], [3], [17]) u E SBV(O), %-'(S,,\K) = 0, the domain SBV( a) contains the original domain of 8. By means of general compactness and lower semicontinuity theorems in SBV( a ) (see [I], [2], [ 31) it can be shown that G has minimizers in SBV( Q). The regularity theorems in [ 13 1, [ 171 show that u E C'(Q\K), , xn-I (s, -n WJ = 0, for any minimizer u so that, by setting K = f7 9, we recover a minimum of 9. The problem of finding effective algorithms for computing the minimizers of G is still widely discussed; see [7], [20], [22], [23], [24], [35]. In this paper we suggest an approach to this problem by approximating in a variational sense, defined by l'-convergence (see, for instance, [ 5 1, [ 121, [ 141, [ 15 I), the functional G by elliptic functionals. The outcome of recent numerical simulations, made by R. March in Pisa, has been very encouraging. Anyway, independently of the possible computational applications of this result, it is interestin...

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In this paper we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X, d, m) which is stable under measured Gromov-Hausdorff convergence and rules out Finsler geometries. It can be given in terms of an enforcement of the Lott, Sturm and Villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow. Besides stability, it enjoys the same tensorization, globalto-local and local-to-global properties. In these spaces, that we call RCD(K, ∞) spaces, we prove that the heat flow (which can be equivalently characterized either as the flow associated to the Dirichlet form, or as the Wasserstein gradient flow of the entropy) satisfies Wasserstein contraction estimates and several regularity properties, in particular Bakry-Emery estimates and the L ∞ − Lip Feller regularization. We also prove that the distance induced by the Dirichlet form coincides with d, that the local energy measure has density given by the square of Cheeger's relaxed slope and, as a consequence, that the underlying Brownian motion has continuous paths. All these results are obtained independently of Poincaré and doubling assumptions on the metric measure structure and therefore apply also to spaces which are not locally compact, as the infinite-dimensional ones.

This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces (X, d, m). Our main results are:• A general study of the relations between the Hopf-Lax semigroup and Hamilton-Jacobi equation in metric spaces (X, d).• The equivalence of the heat flow in L 2 (X, m) generated by a suitable Dirichlet energy and the Wasserstein gradient flow of the relative entropy functional Ent m in the space of probability measures P(X).• The proof of density in energy of Lipschitz functions in the Sobolev space W 1,2 (X, d, m).• A fine and very general analysis of the differentiability properties of a large class of Kantorovich potentials, in connection with the optimal transport problem, is the fourth achievement of the paper.Our results apply in particular to spaces satisfying Ricci curvature bounds in the sense of Lott & Villani [30] and Sturm [39,40] and require neither the doubling property nor the validity of the local Poincaré inequality. MSC-classification: 52C23, 49J52, 49Q20, 58J35, 35K90, 31C25