Catálogo de publicaciones - libros

We prove in this work the trace and trace lifting theorem for Sobolev spaces on the Heisenberg groups for homogeneous hypersurfaces.

We give strong unique continuation and finite jet determination results for mappings between CR manifolds. Applications to the study of groups of local and global CR automorphisms are derived.

We prove that the Cauchy problem for a class of hyperbolic operators with double characteristics and whose simple null bicharacteristics have limit points on the set of double points is not well posed in the C ^∞ category, even though the usual Ivrii-Petkov conditions on the lower order terms are satisfied. According to the standard linear algebra classification these operators, at a double point, have fundamental matrices exhibiting a Jordan block of size 4 and cannot be brought into a canonical form known as “Ivrii decomposition”, due to higher order non-vanishing terms in the Taylor development of the principal symbol near the given double point.

We investigate the possibility of writing f = g ^2 when f is a C ^k nonnegative function with k ≥ 6. We prove that, assuming that f vanishes at all its local minima, it is possible to get g ∈ C ^2 and three times differentiable at every point, but that one cannot ensure any additional regularity.

We prove existence of solutions for the Benjamin—Ono equation with data in H ^s(ℝ), s > 0. Thanks to conservation laws, this yields global solutions for H _1 ^2 (ℝ) data, which is the natural “finite energy” class. Moreover, unconditional uniqueness is obtained in L _t ^∞ ( H _1 ^2 (ℝ)), which includes weak solutions, while for s > _3 ^20 , uniqueness holds in a suitable space.

In [ LPS ], F. Linares, G. Ponce and J.-C. Saut have proved that a non-fully dispersive Zakharov system arising in the study of laser-plasma interaction, is locally well posed in the whole space, for fields vanishing at infinity. Here we show that in the periodic case, seen as a model for fields non-vanishing at infinity, the system develops strong instabilities of Hadamard’s type, implying that the Cauchy problem is strongly ill posed.

We review various classical results on analytic hypoellipticity for operators with double characteristics. Several examples will be discussed to motivate Treves’ conjecture. Finally we announce regularity results obtained recently.

We give sharp regularity conditions, ensuring the backward uniqueness property to a class of parabolic operators.

We present a new approach to the unique determination of the coefficients of the second order hyperbolic equations modulo diffeomorphisms and gauge transformations, assuming that the time-dependent Dirichlet-to-Neumann operator is given on a part of the boundary. We consider also the case of multi-connected domains with obstacles. The interest in this case is spurred by the Aharonov-Bohm effect.

In this paper, we derive sharp observability inequalities for plate equations with lower order terms. More precisely, for any T > 0 and suitable observation domains (satisfying the geometric conditions that the multiplier method imposes), we prove an estimate with an explicit observability constant for plate systems with an arbitrary finite number of components and in any space dimension with lower order bounded potentials. These inequalities are relevant for control theoretical purposes and also in the context of inverse problems. We also prove the optimality of this estimate for plate systems with bounded potentials in even space dimensions n ≥ 2. This is done by extending a construction due to Meshkov to the bi-Laplacian equation, to build a suitable complex-valued bounded potential q = q ( x ), with a non-trivial solution u of Δ ^2 u = qu in ℝ^2, with the decay property | u ( x )| + |∇ u ( x )| + |∇Δ u ( x )| ≤ exp(−| x |^4/3) for all x ∈ ℝ^2.