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The verification of the Nirenberg-Treves conjecture

Nicolas Lerner

Séminaire Bourbaki

In a series of recent papers, Nils Dencker proves that condition ( ψ ) implies the local solvability of principal type pseudodifferential operators (with loss of 3 2 + ϵ derivatives for all positive ϵ ), verifying the last part of the Nirenberg-Treves conjecture, formulated in 1971. The origin of this question goes back to the Hans Lewy counterexample, published in 1957. In this text, we follow the pattern of Dencker’s papers, and we provide a proof of local solvability with a loss of 3 2 derivatives.

Cutting the loss of derivatives for solvability under condition ( Ψ )

Nicolas Lerner — 2006

Bulletin de la Société Mathématique de France

For a principal type pseudodifferential operator, we prove that condition  ( ψ ) implies local solvability with a loss of 3/2 derivatives. We use many elements of Dencker’s paper on the proof of the Nirenberg-Treves conjecture and we provide some improvements of the key energy estimates which allows us to cut the loss of derivatives from ϵ + 3 / 2 for any ϵ > 0 (Dencker’s most recent result) to 3/2 (the present paper). It is already known that condition  ( ψ ) doesimply local solvability with a loss of 1 derivative,...

When is a pseudo-differential equation solvable ?

Nicolas Lerner — 2000

Annales de l'institut Fourier

This paper begins with a broad survey of the state of the art in matters of solvability for differential and pseudo-differential equations. Then we proceed with a Hilbertian lemma which we use to prove a new solvability result.

Transport equations with partially B V velocities

Nicolas Lerner — 2004

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We prove the uniqueness of weak solutions for the Cauchy problem for a class of transport equations whose velocities are partially with bounded variation. Our result deals with the initial value problem t u + X u = f , u | t = 0 = g , where X is the vector fieldwith a boundedness condition on the divergence of each vector field a 1 , a 2 . This model was studied in the paper [LL] with a W 1 , 1 regularity assumption replacing our B V hypothesis. This settles partly a question raised in the paper [Am]. We examine the details of the argument of...

Équations de transport dont les vitesses sont partiellement B V

Nicolas Lerner

Séminaire Équations aux dérivées partielles

Nous démontrons l’unicité des solutions faibles pour une classe d’équations de transport dont les vitesses sont partiellement à variations bornées. Nous nous intéressons à des champs de vecteurs du type a 1 ( x 1 ) · x 1 + a 2 ( x 1 , x 2 ) · x 2 , a 1 B V ( x 1 N 1 ) , a 2 L x 1 1 B V ( x 2 N 2 ) , avec une borne sur la divergence de chacun des champs a 1 , a 2 . Ce modèle a été étudié récemment dans [LL] par C. Le Bris et P.-L. Lions avec une régularité W 1 , 1  ; nous montrons ici également que, dans le cas W 1 , 1 , le contrôle L de...

Lower bounds for pseudo-differential operators

Nicolas LernerJean Nourrigat — 1990

Annales de l'institut Fourier

This paper contains some new results on lower bounds for pseudo-differential operators whose symbols do not remain positive. Non-negativity of averages of the symbol on canonical images of the unit ball is sufficient to get a Gårding type inequality for Schrödinger operators with magnetic potential and one dimensional pseudo-differential operators.

Sharp polynomial energy decay for locally undamped waves

Matthieu LéautaudNicolas Lerner

Séminaire Laurent Schwartz — EDP et applications

In this note, we present the results of the article [LL14], and provide a complete proof in a simple case. We study the decay rate for the energy of solutions of a damped wave equation in a situation where the is violated. We assume that the set of undamped trajectories is a flat torus of positive codimension and that the metric is locally flat around this set. We further assume that the damping function enjoys locally a prescribed homogeneity near the undamped set in traversal directions. We prove...

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