The Monge problem for supercritical Mane potentials on compact manifolds

We prove the existence of an optimal map for the Monge problem when the cost is a supercritical Mane potential on a compact manifold. Supercritical Mane potentials form a class of costs which generalize the Riemannian distances. We describe new links between this transportation problem and viscosity subsolutions of the Hamilton-Jacobi equation

Optimal mass transportation and Mather theory

We study optimal transportation of measures on compact manifolds for costs defined from convex Lagrangians. We prove that optimal transportation can be interpolated by measured Lipschitz laminations, or geometric currents. The methods are inspired from Mather theory on Lagrangian systems. We make use of viscosity solutions of the associated Hamilton-Jacobi equation in the spirit of Fathi's approach to Mather theory

Steady three-dimensional rotational flows: an approach via two stream functions and Nash-Moser iteration

We consider the stationary flow of an inviscid and incompressible fluid of constant density in the region $D=(0, L)\times \mathbb{R}^2$. We are concerned with flows that are periodic in the second and third variables and that have prescribed flux through each point of the boundary $\partial D$. The Bernoulli equation states that the "Bernoulli function" $H:= \frac 1 2 |v|^2+p$ (where $v$ is the velocity field and $p$ the pressure) is constant along stream lines, that is, each particle is associated with a particular value of $H$. We also prescribe the value of $H$ on $\partial D$. The aim of this work is to develop an existence theory near a given constant solution. It relies on writing the velocity field in the form $v=\nabla f\times \nabla g$ and deriving a degenerate nonlinear elliptic system for $f$ and $g$. This system is solved using the Nash-Moser method, as developed for the problem of isometric embeddings of Riemannian manifolds; see e.g. the book by Q. Han and J.-X. Hong (2006). Since we can allow $H$ to be non-constant on $\partial D$, our theory includes three-dimensional flows with non-vanishing vorticity

Weak KAM pairs and Monge-Kantorovich duality

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.Comment: Advanced Studies in Pure Mathematics 47, 2 (2007

Existence and conditional energetic stability of three-dimensional fully localised solitary gravity-capillary water waves

In this paper we show that the hydrodynamic problem for three-dimensional water waves with strong surface-tension effects admits a fully localised solitary wave which decays to the undisturbed state of the water in every horizontal direction. The proof is based upon the classical variational principle that a solitary wave of this type is a critical point of the energy subject to the constraint that the momentum is fixed. We prove the existence of a minimiser of the energy subject to the constraint that the momentum is fixed and small. The existence of a small-amplitude solitary wave is thus assured, and since the energy and momentum are both conserved quantities a standard argument may be used to establish the stability of the set of minimisers as a whole. `Stability' is however understood in a qualified sense due to the lack of a global well-posedness theory for three-dimensional water waves.Comment: 83 pages, 1 figur

Generalized Flows Satisfying Spatial Boundary Conditions

In a region D in $${\mathbb{R}^2}$$ or $${\mathbb{R}^3}$$ , the classical Euler equation for the regular motion of an inviscid and incompressible fluid of constant density is given by $$\partial_t v+(v\cdot \nabla_x)v=-\nabla_x p, {\rm div}_x v=0,$$ where v(t, x) is the velocity of the particle located at $${x\in D}$$ at time t and $${p(t,x)\in\mathbb{R}}$$ is the pressure. Solutions v and p to the Euler equation can be obtained by solving $$\left\{\begin{array}{l} \nabla_x\left\{\partial_t\phi(t,x,a) + p(t,x)+(1/2)|\nabla_x\phi(t,x,a)|^2 \right\}=0\,{\rm at}\,a=\kappa(t,x),\\ v(t,x)=\nabla_x \phi(t,x,a)\,{\rm at}\,a=\kappa(t,x), \\ \partial_t\kappa(t,x)+(v\cdot\nabla_x)\kappa(t,x)=0, \\ {\rm div}_x v(t,x)=0, \end{array}\right. \quad\quad\quad\quad\quad(0.1)$$ where $$\phi:\mathbb{R}\times D\times \mathbb{R}^l\rightarrow\mathbb{R}\,{\rm and}\, \kappa:\mathbb{R}\times D \rightarrow \mathbb{R}^l$$ are additional unknown mappings (l≥ 1 is prescribed). The third equation in the system says that $${\kappa\in\mathbb{R}^l}$$ is convected by the flow and the second one that $${\phi}$$ can be interpreted as some kind of velocity potential. However vorticity is not precluded thanks to the dependence on a. With the additional condition κ(0, x)=x on D (and thus l=2 or 3), this formulation was developed by Brenier (Commun Pure Appl Math 52:411-452, 1999) in his Eulerian-Lagrangian variational approach to the Euler equation. He considered generalized flows that do not cross $${\partial D}$$ and that carry each "particle” at time t=0 at a prescribed location at time t=T>0, that is, κ(T, x) is prescribed in D for all $${x\in D}$$ . We are concerned with flows that are periodic in time and with prescribed flux through each point of the boundary $${\partial D}$$ of the bounded region D (a two- or three-dimensional straight pipe). More precisely, the boundary condition is on the flux through $${\partial D}$$ of particles labelled by each value of κ at each point of $${\partial D}$$ . One of the main novelties is the introduction of a prescribed "generalized” Bernoulli's function $${H:\mathbb{R}^l\rightarrow \mathbb{R}}$$ , namely, we add to (0.1) the requirement that $$\partial_t\phi(t,x,a) +p(t,x)+(1/2)|\nabla_x\phi(t,x,a)|^2=H(a)\,{\rm at}\,a=\kappa(t,x)\quad\quad\quad\quad\quad(0.2)$$ with $${\phi,p,\kappa}$$ periodic in time of prescribed period T>0. Equations (0.1) and (0.2) have a geometrical interpretation that is related to the notions of "Lamb's surfaces” and "isotropic manifolds” in symplectic geometry. They may lead to flows with vorticity. An important advantage of Brenier's formulation and its present adaptation consists in the fact that, under natural hypotheses, a solution in some weak sense always exists (if the boundary conditions are not contradictory). It is found by considering the functional $$(\kappa,v)\rightarrow \int\limits_{0}^T \int\limits_D\left\{\frac 1 2 |v(t,x)|^2+H(\kappa(t,x))\right\}dt\, dx$$ defined for κ and v that are T-periodic in t, such that $$\partial_t\kappa(t,x)+(v\cdot\nabla_x)\kappa(t,x)=0, {\rm div}_x v(t,x)=0,$$ and such that they satisfy the boundary conditions. The domain of this functional is enlarged to some set of vector measures and then a minimizer can be obtained. For stationary planar flows, the approach is compared with the following standard minimization method: to minimize $$\int\limits_{]0,L[\times]0,1[} \{(1/2)|\nabla \psi|^2+H(\psi)\}dx\,{\rm for}\,\psi\in W^{1,2}(]0,L[\times]0,1[)$$ under appropriate boundary conditions, where ψ is the stream function. For a minimizer, corresponding functions $${\phi}$$ and κ are given in terms of the stream function

Discrete spectrum of perturbed Dirac systems with real and periodic coefficients

This paper deals with the number of eigenvalues which appear in the gaps of the spectrum of a Dirac system with real and periodic coefficients when the coefficients are perturbed. The main results provide an upper bound and a condition under which exactly one eigenvalue appears in a given ga