We propose a definition of the Dirichlet energy (which is roughly speaking the integral of the square of the gradient) for mappings mu : Omega -> (P(D), W ) defined over a subset Omega of R^p and valued in the space P(D) of probability measures on a compact convex subset D of R^q endowed with the quadratic Wasserstein distance. Our definition relies on a straightforward generalization of the Benamou-Brenier formula (already introduced by Brenier) but is also equivalent to the definition of Koorevaar, Schoen and Jost as limit of approximate Dirichlet energies, and to the definition of Reshetnyak of Sobolev spaces valued in metric spaces. We study harmonic mappings, i.e. minimizers of the Dirichlet energy provided that the values on the boundary d Omega are fixed. The notion of constant-speed geodesics in the Wasserstein space is recovered by taking for Omega a segment of R. As the Wasserstein space (P(D), W ) is positively curved in the sense of Alexandrov we cannot apply the theory of Koorevaar, Schoen and Jost and we use instead arguments based on optimal transport. We manage to get existence of harmonic mappings provided that the boundary values are Lipschitz on d Omega, uniqueness is an open question. If Omega is a segment of R, it is known that a curve valued in the Wasserstein space P(D) can be seen as a superposition of curves valued in D. We show that it is no longer the case in higher dimensions: a generic mapping Omega -> P(D) cannot be represented as the superposition of mappings Omega -> D. We are able to show a Ishihara-type property: the composition F(mu) of a function F : P(D) -> R convex along generalized geodesics and a harmonic mapping mu : Omega -> P(D) is a subharmonic real-valued function. We also study the special case where we restrict ourselves to a given family of elliptically contoured distributions (a finite-dimensional and geodesically convex submanifold of (P(D), W ) which generalizes the case of Gaussian measures) and show that it boils down to harmonic mappings valued in the Riemannian manifold of symmetric matrices endowed with the distance coming from optimal transport.
In @cite_21 , the authors study (which are nothing more than maps @math except that @math and @math are surfaces, i.e. Riemannian manifolds of dimension @math ) and define a Dirichlet energy in the same way as Koorevaar, Schoen and Jost. These maps are seen as relaxations of "classical" maps @math , and they focus on numerical computation and visualization of theses soft maps, see also @cite_24 for applications to supervised learning. On the other hand, they do not analyze in detail the theoretical properties of the Dirichlet energy and harmonic mappings, which in contrast is the main topic of the present article. In @cite_32 , the author provides some theoretical analysis of soft maps by focusing on the cases where the boundary measures on @math are either Dirac masses or Gaussian measures.
@cite_24 @cite_21 @cite_32
2155488148 2261902652 1886215707 2147668104
Soft maps taking points on one surface to probability distributions on another are attractive for representing surface mappings in the presence of symmetry, ambiguity, and combinatorial complexity. Few techniques, however, are available to measure their continuity and other properties. To this end, we introduce a novel Dirichlet energy for soft maps generalizing the classical map Dirichlet energy, which measures distortion by computing how soft maps transport probabilistic mass from one distribution to another. We formulate the computation of the Dirichlet energy in terms of a differential equation and provide a finite elements discretization that enables all of the quantities introduced to be computed. We demonstrate the effectiveness of our framework for understanding soft maps arising from various sources. Furthermore, we suggest how these energies can be applied to generate continuous soft or point-to-point maps. In this article we consider large energy wave maps in dimension 2+1, as in the resolution of the threshold conjecture by Sterbenz and Tataru (Commun. Math. Phys. 298(1):139–230, 2010; Commun. Math. Phys. 298(1):231–264, 2010), but more specifically into the unit Euclidean sphere ( S ^ n-1 R ^ n ) with ( n ), and study further the dynamics of the sequence of wave maps that are obtained in Sterbenz and Tataru (Commun. Math. Phys. 298(1):231–264, 2010) at the final rescaling for a first, finite or infinite, time singularity. We prove that, on a suitably chosen sequence of time slices at this scaling, there is a decomposition of the map, up to an error with asymptotically vanishing energy, into a decoupled sum of rescaled solitons concentrating in the interior of the light cone and a term having asymptotically vanishing energy dispersion norm, concentrating on the null boundary and converging to a constant locally in the interior of the cone, in the energy space. Similar and stronger results have been recently obtained in the equivariant setting by several authors (Cote, Commun. Pure Appl. Math. 68(11):1946–2004, 2015; Cote, Commun. Pure Appl. Math. 69(4):609–612, 2016; Cote, Am. J. Math. 137(1):139–207, 2015; , Am. J. Math. 137(1):209–250, 2015; Krieger, Commun. Math. Phys. 250(3):507–580, 2004), where better control on the dispersive term concentrating on the null boundary of the cone is provided, and in some cases the asymptotic decomposition is shown to hold for all time. Here, however, we do not impose any symmetry condition on the map itself and our strategy follows the one from bubbling analysis of harmonic maps into spheres in the supercritical regime due to Lin and Riviere (Ann. Math. 149(2):785–829, 1999; Duke Math. J. 111:177–193, 2002), which we make work here in the hyperbolic context of Sterbenz and Tataru (Commun. Math. Phys. 298(1), 231–264, 2010). Wave maps are the simplest wave equations taking their values in a Riemannian manifold (M,g). Their Lagrangian is the same as for the scalar equation, the only difference being that lengths are measured with respect to the metric g. By Noether's theorem, symmetries of the Lagrangian imply conservation laws for wave maps, such as conservation of energy. In coordinates, wave maps are given by a system of semilinear wave equations. Over the past 20 years important methods have emerged which address the problem of local and global wellposedness of this system. Due to weak dispersive effects, wave maps defined on Minkowski spaces of low dimensions, such as @math , present particular technical difficulties. This class of wave maps has the additional important feature of being energy critical, which refers to the fact that the energy scales exactly like the equation. Around 2000 Daniel Tataru and Terence Tao, building on earlier work of Klainerman–Machedon, proved that smooth data of small energy lead to global smooth solutions for wave maps from 2+1 dimensions into target manifolds satisfying some natural conditions. In contrast, for large data, singularities may occur in finite time for @math as target. This monograph establishes that for @math as target the wave map evolution of any smooth data exists globally as a smooth function. While we restrict ourselves to the hyperbolic plane as target the implementation of the concentration-compactness method, the most challenging piece of this exposition, yields more detailed information on the solution. This monograph will be of interest to experts in nonlinear dispersive equations, in particular to those working on geometric evolution equations. We show that the Goemans-Linial semidefinite relaxation of the Sparsest Cut problem with general demands has integrality gap @math . This is achieved by exhibiting @math -point metric spaces of negative type whose @math distortion is @math . Our result is based on quantitative bounds on the rate of degeneration of Lipschitz maps from the Heisenberg group to @math when restricted to cosets of the center.