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A classical problem of transporting mass due to Monge and Kantorovich is solved. Given measures µ and ν on R d, we find the measure-preserving map y(x) between them with minimal cost — where cost is measured against h(x − y) withhstrictly convex, or a strictly concave function of |x − y|. This map is unique: it is characterized by the formula y(x) =x−(∇h) −1 (∇ψ(x)) and geometrical restrictions on ψ. Connections with mathematical economics, numerical computations, and the Monge-Ampère equation are sketched. ∗ Both authors gratefully acknowledge the support provided by postdoctoral fellowships: WG at

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The subdifferential of a lower semicontinuous proper convex function on a Banach space is a maximal monotone operator, as well as a maximal cyclically monotone operator. This result was announced by the author in a previous paper, but the argument given there was incomplete; the result is proved here by a different method, which is simpler in the case of reflexive Banach spaces. At the same time, a new fact is established about the relationship between the subdifferential of a convex function and the subdifferential of its conjugate in the nonreflexive case. Let E be a real Banach space with dual E*. A proper convex function on E is a function f from E to (- 00, + 00 J, not identically + (0, such that f«l- J\,)x+ J\,y) ~ (1- J\,)f(x) + J\,f(y) whenever x E E, y E E and 0 < J\, < 1. The subdiffereniiai of such a function f is the (generally multivalued) mapping of: E-> E * defined by of (x) = {x * E E * I f(y) ~ f(x) + <y- x, x*>, Vy E E}, where <.,.> denotes the canonical pairing between E and E*. A multivalued mapping T: E-> E * is said to be a monotone operator if It is said to be a cyclically monotone operator if

Recently Clarke, Stern and Wolenski characterized, in a Hilbert space, the closed subsets C for which the distance function dC is continuously differentiable everywhere on an open “tube” of uniform thickness around C. Here a corresponding local theory is developed for the property of dC being continuously differentiable outside of C on some neighborhood of a point x ∈ C. This is shown to be equivalent to the prox-regularity of C at x, which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable through a calculus. Additional characterizations are provided in terms of d 2 C being locally of class C1+ or such that d 2 C + σ | · |2 is convex around x for some σ> 0. Prox-regularity of C at x corresponds further to the normal cone mapping NC having a hypomonotone truncation around x and leads to a formula for PC by way of NC. The local theory also yields new insights on the global level of the Clarke-Stern-Wolenski results and on a property of sets introduced by Shapiro, as well as on the concept of sets with positive reach considered by Federer in finite dimensions.

We study the backward Euler method with variable time-steps for abstract evolution equations in Hilbert spaces. Exploiting convexity of the underlying potential or the angle-bounded condition, thereby assuming no further regularity, we derive novel a posteriori estimates of the discretization error in terms of computable quantities related to the amount of energy dissipation or monotonicity residual. These estimators solely depend on the discrete solution and data and impose no constraints between consecutive time-steps. We also prove that they converge to zero with an optimal rate with respect to the regularity of the solution. We apply the abstract results to a number of concrete strongly nonlinear problems of parabolic type with degenerate or singular character.

We consider the method for constrained convex optimization in a Hilbert space, consisting of a step in the direction opposite to an ek-subgradient of the objective at a current iterate, followed by an orthogonal projection onto the feasible set. The normalized stepsizes ek are exogenously given, satisfying ~=0 c~k ec, ~=0 c ~ < ec, and ek is chosen so that ek ~</~k for some #> 0. We prove that the sequence generated in this way is weakly convergent to a minimizer if the problem has solutions, and is unbounded otherwise. Among the features of our convergence analysis, we mention that it covers the nonsmooth case, in the sense that we make no assumption of differentiability off, and much less of Lipschitz continuity of its gradient. Also, we prove weak convergence of the whole sequence, rather than just boundedness of the sequence and optimality of its weak accumulation points, thus improving over all previously known convergence results. We present also convergence rate results. © 1998 The Mathematical

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We study the flow of two immiscible fluids of different density and mobility in a porous medium. If the heavier phase lies above the lighter one, the interface is observed to be unstable. The two phases start to mix on a mesoscopic scale and the mixing zone grows in time --- an example of evolution of microstructure. A simple set of assumptions on the physics of this two--phase flow in a porous medium leads to a mathematically ill--posed problem --- when used to establish a continuum free boundary problem. We propose and motivate a relaxation of this "nonconvex" constraint of a phase distribution with a sharp interface on a macroscopic scale. We prove that this approach leads to a mathematically well--posed problem which predicts shape and evolution of the mixing profile as a function of the density difference and mobility quotient. 1 Introduction We are interested in the flow of two immiscible fluids of different density and mobility in a porous medium. If the more dense phase lies a...

Maximal monotone operator theory is about to turn (or just has turned) fifty. I intend to briefly survey the history of the subject. I shall try to explain why maximal monotone operators are both interesting and important—culminating with a description of the remarkable progress made during the past decade. 1