Search results for "Elliptic operator"

On essential maximality of linear pseudo-differential operators

1989

Partial {$*$}-algebras of closable operators. I. The basic theory and the abelian case

1990

This paper, the first of two, is devoted to a systematic study of partial *-algebras of closable operators in a Hilbert space (partial Op*-algebras). After setting up the basic definitions, we describe canonical extensions of partial Op*-algebras by closure and introduce a new bounded commutant, called quasi-weak. We initiate a theory of abelian partial *-algebras. As an application, we analyze thoroughly the partial Op*-algebras generated by a single closed symmetric operator.

Sur une classe d’equations du type parabolique lineaires

1996

The application of the variational method for the existence theorem, developped by J. L. Lions, for the evolution equations in Hilbert spaces to a considerably large class of systems of linear partial differential equations of parabolic type is studied by defining Hilbert spaces in relation to the elliptic operator of the system, and an example insired by the system of equations for a viscous gas is examined.

On Determinants of Integrable Operators with Shifts

2013

Integrable integral operator can be studied by means of a matrix Riemann--Hilbert problem. However, in the case of so-called integrable operators with shifts, the associated Riemann--Hilbert problem becomes operator valued and this complicates strongly the analysis. In this note, we show how to circumvent, in a very simple way, the use of such a setting while still being able to characterize the large-$x$ asymptotic behavior of the determinant associated with the operator.

Explicit form of the time operator of a gaussian stationary process

2004

We present the time operator theory in the framework of stationary stochastic processes. The main results of the paper is the derivation of the time operator acting on the Fock space associated with a discrete time gaussian stationary process.

2003

In this article we apply the S(M, g)–calculus of L. Hormander and, in particular, results concerning the spectral invariance of the algebra of operators of order zero in ℒ(L2(ℝn)) to study generators of Feller semigroups. The core of the article is the proof of the invertibility of λ Id + P for a strongly elliptic operator P in Ψ(M, g) and suitable weight functions M and metrics g. The proof depends highly on precise estimates of the remainder term in asymptotic expansions of the product symbol in Weyl and Kohn–Nirenberg quantization. Due to the Hille–Yosida–Ray theorem and a theorem of Courrege, the result concerning the invertibility of λ Id + P is applicable to obtain sufficient conditio…

Zero viscosity limit of the Oseen equations in a channel

2001

Oseen equations in the channel are considered. We give an explicit solution formula in terms of the inverse heat operators and of projection operators. This solution formula is used for the analysis of the behavior of the Oseen equations in the zero viscosity limit. We prove that the solution of Oseen equations converges in W1,2 to the solution of the linearized Euler equations outside the boundary layer and to the solution of the linearized Prandtl equations inside the boundary layer. © 2001 Society for Industrial and Applied Mathematics.

Spherical harmonic expansion of fundamental solutions and their derivatives for homogenous elliptic operators

2017

In this work, a unified scheme for computing the fundamental solutions of a three-dimensional homogeneous elliptic partial differential operator is presented. The scheme is based on the Rayleigh expansion and on the Fourier representation of a homogeneous function. The scheme has the advantage of expressing the fundamental solutions and their derivatives up to the desired order without any term-by-term differentiation. Moreover, the coefficients of the series need to be computed only once, thus making the presented scheme attractive for numerical implementation. The scheme is employed to compute the fundamental solution of isotropic elasticity showing that the spherical harmonics expansions…

Quantitative Runge Approximation and Inverse Problems

2017

In this short note we provide a quantitative version of the classical Runge approximation property for second order elliptic operators. This relies on quantitative unique continuation results and duality arguments. We show that these estimates are essentially optimal. As a model application we provide a new proof of the result from \cite{F07}, \cite{AK12} on stability for the Calder\'on problem with local data.

Distribution of Large Eigenvalues for Elliptic Operators

2019

In this chapter we consider elliptic differential operators on a compact manifold and rather than taking the semi-classical limit (h →), we let h = 1 and study the distribution of large eigenvalues. Bordeaux Montrieux (Loi de Weyl presque sure et resolvante pour des operateurs differentiels non-autoadjoints, these, CMLS, Ecole Polytechnique, 2008. https://pastel.archives-ouvertes.fr/pastel-00005367, Ann Henri Poincare 12:173–204, 2011) studied elliptic systems of differential operators on S1 with random perturbations of the coefficients, and under some additional assumptions, he showed that the large eigenvalues obey the Weyl law almost surely. His analysis was based on a reduction to the s…