We study the Monge transportation problem when the cost is the action associated to a Lagrangian function on a compact manifold. We show that the transportation can be interpolated by a Lipschitz lamination. We describe several direct variational problems the minimizers of which are these Lipschitz laminations. We prove the existence of an optimal transport map when the transported measure is absolutely continuous. We explain the relations with Mather’s minimal measures.

Abstract. Here shape space is either the manifold of simple closed smooth unparameterized curves in R 2 or is the orbifold of immersions from S 1 to R 2 modulo the group of diffeomorphisms of S 1. We investige several Riemannian metrics on shape space: L 2-metrics weighted by expressions in length and curvature. These include a scale invariant metric and a Wasserstein type metric which is sandwiched between two length-weighted metrics. Sobolev metrics of order n on curves are described. Here the horizontal projection of a tangent field is given by a pseudo-differential operator. Finally the metric induced from the Sobolev metric on the group of diffeomorphisms on R 2 is treated. Although the quotient metrics are all given by pseudo-differential operators, their inverses are given by convolution with smooth kernels. We are able to prove local existence and uniqueness of solution to the geodesic equation for both kinds of Sobolev metrics. We are interested in all conserved quantities, so the paper starts with the Hamiltonian setting and computes conserved momenta and geodesics in general on the space of immersions. For each metric we compute the geodesic equation on shape space. In the end we sketch in some examples the differences between these metrics.

Abstract. We survey recent advances in analysis and geometry, where first order differential analysis has been extended beyond its classical smooth settings. Such studies have applications to geometric rigidity questions, but are also of intrinsic interest. The transition from smooth spaces to singular spaces where calculus is possible parallels the classical development from smooth functions to functions with weak or generalized derivatives. Moreover, there is a new way of looking at the classical geometric theory of Sobolev functions that is useful in more general contexts. 1.

Dedicated to Ivo Babuˇska on the occasion of his 80th birthday We present a variational framework for shape optimization problems that establishes clear and explicit connections among the continuous formulation, its full discretization and the resulting linear algebraic systems. Our approach hinges on the following essential features: shape differential calculus, a semi-implicit time discretization and a finite element method for space discretization. We use shape differential calculus to express variations of bulk and surface energies with respect to domain changes. The semi-implicit time discretization allows us to track the domain boundary without an explicit parametrization, and has the flexibility to choose different descent directions by varying the scalar product used for the computation of normal velocity. We propose a Schur complement approach to solve the resulting linear systems efficiently. We discuss applications of this framework to image segmentation, optimal shape design for PDE,

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We consider the asymptotic behavior of radially symmetric solutions of the aggregation equation ut = ∇ · (u∇K ∗ u) in R n, for homogeneous potentials K = |x | γ, γ> 0. For γ> 2, the aggregation happens in infinite time and exhibits a concentration of mass along a collapsing δ-ring. We develop an asymptotic theory for the approach to this singular solution. For γ < 2, the solution blows up in finite time and we present careful numerics of second type similarity solutions for all γ in this range, including additional asymptotic behavior in the limits γ → 0 + and γ → 2 −.

Abstract. Let a smooth family of Riemannian metrics g(τ) satisfy the backwards Ricci flow equation on a compact oriented n-dimensional manifold M. Suppose two families of normalized n-forms ω(τ) ≥ 0 and ˜ω(τ) ≥ 0 satisfy the forwards (in τ) heat equation on M generated by the connection Laplacian ∆g(τ). If these n-forms represent two evolving distributions of particles over M, the minimum root-mean-square distance W2(ω(τ), ˜ω(τ), τ) to transport the particles of ω(τ) onto those of ˜ω(τ) is shown to be non-increasing as a function of τ, without sign conditions on the curvature of (M, g(τ)). Moreover, this contractivity property is shown to characterize supersolutions to the Ricci flow.

The dynamics of globally minimizing orbits of Lagrangian systems can be studied using the Barrier function, as Mather first did, or using the pairs of weak KAM solutions introduced by Fathi. The central observation of the present paper is that Fathi weak KAM pairs are precisely the admissible pairs for the Kantorovich problem dual to the Monge transportation problem with the Barrier function as cost. We exploit this observation to recover several relations between the Barrier functions and the set of weak KAM pairs in an axiomatic and elementary way. Let M be a compact connected manifold and consider a C 2 Lagrangian function L: TM × R → R that satisfies the standard hypotheses of the calculus of variations,

We consider a pattern-forming system in two space dimensions defined by an energy G ". The functional G " models strong phase separation in AB diblock copolymer melts, and patterns are represented by f0; 1g-valued functions; the values 0 and 1 correspond to the A and B phases. The parameter " is the ratio between the intrinsic, material length-scale and the scale of the domain. We show that in the limit " ! 0 any sequence u " of patterns with uniformly bounded energy G "ðu"Þ becomes stripe-like: the pattern becomes locally one-dimensional and resembles a periodic stripe pattern of periodicity Oð"Þ. In the limit the stripes become uniform in width and increasingly straight. Our results are formulated as a convergence theorem, which states that the functional G " Gamma-converges to a limit functional G0. This limit functional is defined on fields of rank-one projections, which represent the local direction of the stripe pattern. The functional G0 is only finite if the projection field solves a version of the Eikonal equation, and in that case it is the L2-norm of the divergence of the projection field, or equivalently the L2-norm of the curvature of the field. At the level of patterns the converging objects are the jump measures jru "j combined with the projection fields corresponding to the tangents to the jump set. The central inequality from Peletier and R€oger, Arch. Rational Mech. Anal. 193 (2009) 475 537, provides the initial estimate and leads to weak measure-function pair convergence. We obtain strong convergence by exploiting the non-intersection property of the jump set.

We consider the coarse-graining of a lattice system with continuous spin variable. In the first part, two abstract results are established: sufficient conditions for a logarithmic Sobolev inequality with constants independent of the dimension (Theorem 3) and sufficient conditions for convergence to the hydrodynamic limit (Theorem 8). In the second part, we use the abstract results to treat a specific example, namely the Kawasaki dynamics with Ginzburg–