On considère le problème mixte dans un quadrant pour un opérateur différentiel hyperbolique $P$ en supposant que $P$ et les opérateurs au bord sont homogènes à coefficients constants. On caractérise les conditions au bord pour avoir existence et unicité de la solution du problème mixte, en se plaçant successivement dans le cadre des fonctions $C^{\infty }$, puis, lorsque $P$ est strictement hyperbolique, dans le cadre des espaces de Sobolev. Ces caractérisations s’expriment au moyen d’une condition dite de Lopatinski,...

Nous donnons une condition suffisante pour qu’un opérateur de Schrödinger avec champ magnétique soit à résolvante compacte. Dans le cas où cette condition n’est pas verifiée, nous caractérisons son spectre essentiel.

We consider the Darboux problem for a functional differential equation: $\left(\partial ²u\right)/\left(\partial x\partial y\right)(x,y)=f(x,y,u_{(x,y)},u(x,y),\partial u/\partial x(x,y),\partial u/\partial y(x,y))$ a.e. in [0,a]×[0,b], u(x,y) = ψ(x,y) on [-a₀,a]×[-b₀,b]∖(0,a]×(0,b], where the function $u_{(x,y)}:[-a₀,0]\times [-b₀,0]\to {ℝ}^k$ is defined by $u_{(x,y)}(s,t)=u(s+x,t+y)$ for (s,t) ∈ [-a₀,0]×[-b₀,0]. We give a few theorems about weak and strong inequalities for this problem. We also discuss the case where the right-hand side of the differential equation is linear.

In this paper we complete the characterization of those $f$, $\mu$ and $\nu$ such that $${\parallel w\parallel }_{H^1(\mathrm{\Omega })}^2+{\int }_{B}f(x,w(x))d\mu +\nu (B)$$ is $\mathrm{\Gamma }(L^2{(\mathrm{\Omega })}^{-})$ limit of a sequence of obstacles ${\parallel w\parallel }_{H^1(\mathrm{\Omega })}^2+{\mathrm{\Phi }}_h(w,B)$ where

We establish a Carleman type inequality for the subelliptic operator $ℒ={\Delta }_z+|x{|}^2{\partial }_t^2$ in ${ℝ}^{n+1}$, $n\ge 2$, where $z\in {ℝ}^n$, $t\in ℝ$. As a consequence, we show that $-ℒ+V$ has the strong unique continuation property at points of the degeneracy manifold $\left\{\right(0,t)\in {ℝ}^{n+1}|t\in ℝ\}$ if the potential $V$ is locally in certain $L^p$ spaces.

In this paper, we establish Carleman estimates for the two dimensional isotropic non-stationary Lamé system with the zero Dirichlet boundary conditions. Using this estimate, we prove the uniqueness and the stability in determining spatially varying density and two Lamé coefficients by a single measurement of solution over $(0,T)\times \omega$, where $T\>0$ is a sufficiently large time interval and a subdomain $\omega$ satisfies a non-trapping condition.

In this paper, we establish Carleman estimates for the two dimensional isotropic non-stationary Lamé system with the zero Dirichlet boundary conditions. Using this estimate, we prove the uniqueness and the stability in determining spatially varying density and two Lamé coefficients by a single measurement of solution over (0,T) x ω, where T > 0 is a sufficiently large time interval and a subdomain ω satisfies a non-trapping condition.

We derive Carleman type estimates with two large parameters for a general partial differential operator of second order. The weight function is assumed to be pseudo-convex with respect to the operator. We give applications to uniqueness and stability of the continuation of solutions and identification of coefficients for the Lamé system of dynamical elasticity with residual stress. This system is anisotropic and cannot be principally diagonalized, but it can be transformed into an "upper triangular"...

This article is devoted to incompressible Euler equations (or to Navier-Stokes equations in the vanishing viscosity limit). It describes the propagation of quasi-singularities. The underlying phenomena are consistent with the notion of a cascade of energy.

This paper outlines the manner in which Thom’s theory of catastrophes fits into the Hamilton-Jacobi theory of partial differential equations. The representation of solutions of a first order partial differential equation as lagrangian manifolds allows one to study the local structure of their singularities. The structure of generic singularities is closely related to Thom’s concept of the elementary catastrophe associated to a singularity. Three concepts of the stability of a singularity are discussed....