Abstract. We define distances between geometric curves by the square root of the minimal energy required to transform one curve into the other. The energy is formally defined from a left invariant Riemannian distance on an infinite dimensional group acting on the curves, which can be explicitly computed. The obtained distance boils down to a variational problem for which an optimal matching between the curves has to be computed. An analysis of the distance when the curves are polygonal leads to a numerical procedure for the solution of the variational problem, which can efficiently be implemented, as illustrated by experiments.

This paper proves the estimate ku" uk L 1 (Q T ) C" 1=5 , where, for all " > 0, u" is the weak solution of (u" ) t + div(q f(u" )) ('(u")+"u") = 0 with initial and boundary conditions, u is the entropy weak solution of u t + div(qf(u)) ('(u)) = 0 with the same initial and boundary conditions, and C > 0 does not depend on ". The domain

We present a new mathematical formulation of some curve and surface reconstruction algorithms by the introduction of auxiliary variables. For deformable models and templates, the extraction of a shape is obtained through the minimization of an energy composed of an internal regularization term (not necessary in the case of parametric models) and an external attraction potential. Two-step iterative algorithms have been often used where, at each iteration, the model is first locally deformed according to the potential data attraction and then globally smoothed (or fitted in the parametric case). We show how these approaches can be interpreted as the introduction of auxiliary variables and the minimization of a two-variables energy. The first variable corresponds to the original model we are looking for, while the second variable represents an auxiliary shape close to the first one. This permits to transform an implicit data constraint defined by a non convex potential into an explicit co...

One approximates the entropy weak solution u of a nonlinear parabolic degenerate equation u t + div(qf(u)) '(u) = 0 by a piecewise constant function uD using a discretization D in space and time and a finite volume scheme. The convergence of uD to u is shown as the size of the space and time steps tend to zero. In a first step, estimates on uD are used to prove the convergence, up to a subsequence, of uD to a measure valued entropy solution (called here an entropy process solution). A result of uniqueness of the entropy process solution is proved, yielding the strong convergence of uD to u. Some numerical results on a model equation are shown.

Part I. The aim of this paper is to study the tangential trace and tangential components of fields which belong to the space H(curl ;\Omega\Gamma2 when\Omega is a polyhedron with Lipschitz continuous boundary. The appropriate functional setting is developed in order to suitably define these traces on the whole boundary and on a part of it (for partially vanishing fields and general ones.) In both cases it is possible to define ad hoc dualities among tangential trace and tangential components and the validity of two related integration by parts formulae is provided. Part II. Hodge decompositions of tangential vector fields defined on piecewise regular manifolds are provided. The first step is the study of L 2 tangential fields and then the attention is focused on some particular Sobolev spaces of order \Gamma1=2. In order to reach this goal, it is required to properly define the first order differential operators and to study their properties. When the manifold \Gamma is the boundar...

Payne-Pólya-Weinberger conjecture, Sperner’s inequality, biharmonic operator, bi-Laplacian, clamped plate problem, Rayleigh’s conjecture, buckling problem, the Pólya-Szegő conjecture, universal inequalities for eigenvalues, Hile-Protter inequality, H. C. Yang’s inequality. Short title: Isoperimetric and Universal Inequalities This paper reviews many of the known inequalities for the eigenvalues of the Laplacian and bi-Laplacian on bounded domains in Euclidean space. In particular, we focus on isoperimetric inequalities for the low eigenvalues of the Dirichlet and Neumann Laplacians and of the vibrating clamped plate problem (i.e., the biharmonic operator with “Dirichlet ” boundary conditions). We also discuss the known universal inequalities for the eigenvalues of the Dirichlet Laplacian and the vibrating clamped plate and buckling problems and go on to

In this paper, we discuss a geometrical model of a space of deformable images or shapes, in which infinitesimal variations are combinations of elastic deformations (warping) and of photometric variations. Geodesics in this space are related to velocity-based image warping methods, which have proved to yield efficient and robust estimations of diffeomorphisms in the case of large deformation. Here, we provide a rigorous and general construction of this infinite dimensional “shape manifold ” on which we place a Riemannian metric. We then obtain the geodesic equations, for which we show the existence and uniqueness of solutions for all times. We finally use this to provide a geometrically founded linear approximation of the deformations of shapes in the neighborhood of a given template.

Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the Laplace-Beltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and

A general abstract duality result is proposed for equations which...

Abstract not found