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A parametrix construction for wave equations with C 1 , 1 coefficients

Hart F. Smith (1998)

Annales de l'institut Fourier

In this article we give a construction of the wave group for variable coefficient, time dependent wave equations, under the hypothesis that the coefficients of the principal term possess two bounded derivatives in the spatial variables, and one bounded derivative in the time variable. We use this construction to establish the Strichartz and Pecher estimates for solutions to the Cauchy problem for such equations, in space dimensions n = 2 and n = 3 .

An accuracy improvement in Egorov's theorem.

Jorge Drumond Silva (2007)

Publicacions Matemàtiques

We prove that the theorem of Egorov, on the canonical transformation of symbols of pseudodifferential operators conjugated by Fourier integral operators, can be sharpened. The main result is that the statement of Egorov's theorem remains true if, instead of just considering the principal symbols in Sm/Sm-1 for the pseudodifferential operators, one uses refined principal symbols in Sm/Sm-2, which for classical operators correspond simply to the principal plus the subprincipal symbol, and can generally...

Boundedness of Fourier integral operators on Fourier Lebesgue spaces and affine fibrations

Fabio Nicola (2010)

Studia Mathematica

We study Fourier integral operators of Hörmander’s type acting on the spaces L p ( d ) c o m p , 1 ≤ p ≤ ∞, of compactly supported distributions whose Fourier transform is in L p . We show that the sharp loss of derivatives for such an operator to be bounded on these spaces is related to the rank r of the Hessian of the phase Φ(x,η) with respect to the space variables x. Indeed, we show that operators of order m = -r|1/2-1/p| are bounded on L p ( d ) c o m p if the mapping x x Φ ( x , η ) is constant on the fibres, of codimension r, of an affine...

Continuity of solutions of a quasilinear hyperbolic equation with hysteresis

Petra Kordulová (2012)

Applications of Mathematics

This paper is devoted to the investigation of quasilinear hyperbolic equations of first order with convex and nonconvex hysteresis operator. It is shown that in the nonconvex case the equation, whose nonlinearity is caused by the hysteresis term, has properties analogous to the quasilinear hyperbolic equation of first order. Hysteresis is represented by a functional describing adsorption and desorption on the particles of the substance. An existence result is achieved by using an approximation of...

Déterminants et intégrales de Fresnel

Yves Colin de Verdière (1999)

Annales de l'institut Fourier

On présente ici une approche directe et géométrique pour le calcul des déterminants d’opérateurs de type Schrödinger sur un graphe fini. Du calcul de l’intégrale de Fresnel associée, on déduit le déterminant. Le calcul des intégrales de Fresnel est grandement facilité par l’utilisation simultanée du théorème de Fubini et d’une version linéaire du calcul symbolique des opérateurs intégraux de Fourier. On obtient de façon directe une formule générale exprimant le déterminant en terme des conditions...

Dynamics of wave propagation and curvature of discriminants

Victor P. Palamodov (2000)

Annales de l'institut Fourier

For a Lagrange distribution of order zero we consider a quadratic integral which has logarithmic divergence at the singular locus of the distribution. The residue of the asymptotics is a Hermitian form evaluated in the space of positive distributions supported in the locus. An asymptotic analysis of the residue density is given in terms of the curvature form of the locus. We state a conservation law for the residue of the impulse-energy tensor of solutions of the wave equation which extends the...

Eigenvalue distribution for non-self-adjoint operators with small multiplicative random perturbations

Johannes Sjöstrand (2009)

Annales de la faculté des sciences de Toulouse Mathématiques

In this work we continue the study of the Weyl asymptotics of the distribution of eigenvalues of non-self-adjoint (pseudo)differential operators with small random perturbations, by treating the case of multiplicative perturbations in arbitrary dimension. We were led to quite essential improvements of many of the probabilistic aspects.

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