Nicolas Lerner

Hyperbolic operators with non-Lipschitz coefficients

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Uniqueness of continuous solutions for BV vector fields

We consider a vector field whose coefficients are functions of bounded variation, with a bounded divergence. We prove the uniqueness of continuous solutions for the Cauchy problem.

The Wick calculus of pseudo-differential operators and energy estimates

Acknowledgement. This is my pleasure to congratulate Professor Komatsu on his sixtieth birthday and to thank him for his invitation and the very warm welcome he gave us during our stay in Japan. I wish also to express my thanks to the other organizers of this meeting, Professors Bony and Tose.

Geometrical Optics and Related Topics

Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions - an outline of the proof, S. Alinhac concentration effects in critical nonlinear wave equation and scattering theory, H. Bahouri and P. Gerard lower semicontinuity of weighted path length in BV, P. Baiti and A. Bressan time decay of Lp norms for solutions of the wave equation on exterior domains, M. Beals Sobolev embeddings in Weyl-Hormander calculus, J-Y. Chemin and C-J. Xu about the Cauchy problem for a system of conservation laws, C. Cheverry global existance of the solutions and formation of singularities for a class of hyberbolic systems, D. DeL Santo, V. Georgiev and E. Mitidieri a class of solvable operators, N. Dencker uniqueness of the Cauchy problem under partial analyticity assumptions, L. Hormander nonlinear wave diffraction, J.K. Hunter caustics for dissipative semilinear oscillations, J-L. Joly, G. Metivier and J. Rauch geometric optics and the bottom of the spectrum, R.B. Melrose hypoellipticity for a class of infinately degenerate elliptic operators, Y. Morimoto and T. Morioka regularity of solutions to characteristic boundary value problem for symmetric systems, T. Nishitani and M. Takayama.

TRACE THEOREMS FOR PSEUDO-DIFFERENTIAL OPERATORS

An integral operator with smooth kernel can always be restricted to a hypersurfaceS. Acutally, it is again an integral operator and its kernel is the restriction (in both variables) of the original one toS. Here we study restrictions of pseudo-differential operators of arbitrary order. We find sufficient and (to some extent) necessary conditions on the symbol ensuring existence of the restriction. These conditions require the vanishing of some geometrical invariants defined on the conormal bundle of the hypersurface. In particular, for a pseudo-differential operator of orderm, the principal symbol should vanish of order [m]+2 and the subprincipal symbol of order [m]+1. These classical invariants are sufficient to treat the problem for the casem<1, but in the general case we need to introduce new higher order invariants related to the operator and the hypersurface.

Perturbation and energy estimates

Abstract We prove in this paper that given a first order pseudo-differential operator P satisfying Nirenberg-Treves condition (ψ), there exists an L 2 -bounded operator R so that P + R is solvable. Solvability occurs with the loss of two derivatives. We prove along the way a natural factorization result for operators satisfying condition (ψ).

Propagation des singularités Gevrey pour l’équation de Cauchy-Riemann dégénérée

In this paper we study the degenerate Cauchy-Riemann equation in Gevrey classes. We first prove the local solvability in Gevrey classes of functions and ultra-distributions. Using microlocal techniques with Fourier integral operators of infinite order and microlocal energy estimates, we prove a result of propagation of singularities along one dimensional bicharacteristics.

Second Microlocalization Methods for Degenerate Cauchy—Riemann Equations

We study a class of degenerate Cauchy—Riemann equations and we show that the second microlocalization with respect to a hypersurface is a useful tool to formulate and prove propagation and solvability results.

A Gårding Inequality on a Manifold with Boundary

Let a ∈ C∞ (ℝ n × ℝ n ) and m ∈ ℝ. The function a is said to be a symbol of order m if one has estimates