The spectrum of an elliptic operator of second order

Under minimal requirements on the coefficients and the boundary of the domain it is proved that the spectrum of the first boundary-value problem for an elliptic operator of second order always lies in the half-plane Re , where is the leading eigenvalue to which there corresponds a nonnegative eigenfunction. On the line Re = , there are no other points of the spectrum.Translated from Matematicheskie Zametki, Vol. 20, No. 3, pp. 351–358, September, 1976.     

Bound for resolvent of second order elliptic operator in low spectral stripe

Mourre method of commutators is used to get low energy resolvent bound for an abstract operatorin a Hilbert space and for a second order variable coefficient elliptic operator in R^d, d?3. Copyright © 2002 John Wiley & Sons, Ltd.     

Concentration of the first eigenfunction for a second order elliptic operator

We study the semi-classical limits of the first eigenfunction of a positive second order operator on a compact Riemannian manifold when the diffusion constant ε goes to zero. We assume that the first order term is given by a vector field b, whose recurrent components are either hyperbolic points or cycles or two dimensional torii. The limits of the normalized eigenfunctions concentrate on the recurrent sets of maximal dimension where the topological pressure [Y. Kifer, Principal eigenvalues, topological pressure and stochastic stability of equilibrium states, Israel J. Math. 70 (1990) (1) 1–47] is attained. On the cycles and torii, the limit measures are absolutely continuous with respect to the invariant probability measure on these sets. We have determined these limit measures, using a blow-up analysis. To cite this article: D. Holcman, I. Kupka, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

ESTIMATES OF EIGENVALUES FOR UNIFORMLY ELLIPTIC OPERATOR OF SECOND ORDER

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Asymptotics of the parabolic Green function for an elliptic operator on a manifold with conical points

We construct the asymptotics ast→0 of the trace of the operator exp(−tP) for an elliptic operatorP on a manifold with conical points. Translated fromMatematicheskie Zametki, Vol. 63, No. 1, pp. 28–36, January, 1998. This research was supported by the Belarus Foundation for Basic Research under grant No. 96-01 00790.     

Asymptotics of the parabolic Green function for an elliptic operator on a manifold with conical points

We construct the asymptotics ast→0 of the trace of the operator exp(?tP) for an elliptic operatorP on a manifold with conical points.     

Linear differential operators and operator matrices of the second order

Linear differential operators (equations) of the second order in Banach spaces of vector functions defined on the entire real axis are studied. Conditions of their invertibility are given. The main results are based on putting a differential operator in correspondence with a second-order operator matrix and further use of the theory of first-order differential operators that are defined by the operator matrix. A general scheme is presented for studying the solvability conditions for different classes of second-order equations using second-order operator matrices. The scheme includes the studied problem as a special case.     

Markovian extensions of symmetric second order elliptic differential operators

Let be bounded with a smooth boundary Γ and let S be the symmetric operator in given by the minimal realization of a second order elliptic differential operator. We give a complete classification of the Markovian self‐adjoint extensions of S by providing an explicit one‐to‐one correspondence between such extensions and the class of Dirichlet forms in which are additively decomposable by the bilinear form of the Dirichlet‐to‐Neumann operator plus a Markovian form. By such a result two further equivalent classifications are provided: the first one is expressed in terms of an additive decomposition of the bilinear forms associated to the extensions, the second one uses the additive decomposition of the resolvents provided by Kre?n's formula. The Markovian part of the decomposition allows to characterize the operator domain of the corresponding extension in terms of Wentzell‐type boundary conditions. Some properties of the extensions, and of the corresponding Dirichlet forms, semigroups and heat kernels, like locality, regularity, irreducibility, recurrence, transience, ultracontractivity and Gaussian bounds are also discussed.     

Asymptotics of the spectrum of a second-order nonselfadjoint degenerate elliptic differential operator on the interval

Some properties are studied of a degenerate elliptic operator P defined on the interval (0, 1); namely, the resolvent of P is estimated. The completeness is investigated of the system of vector functions of P, and the summability is studied by the Abel method with parentheses of the Fourier series of elements in the corresponding Hilbert spaces with respect to systems of the root vector functions of P. An asymtotic formula is obtained for the distribution of the eigenvalues of P that distinguishes the principal term of the asymptotics.     

Functions of a second order elliptic operator in rearrangement invariant spaces

The functional calculus of positive operators is applied to second-order elliptic operatorsP. For any absolutely concave (t), the corresponding operators (P ^–1) are represented as integral operators, their kernels are estimated, and these estimates are used for studying (P ^–1) in Lorentz, Marcinkiewicz and Orlicz spaces. Most of results obtained are sharp.     

Measure algebras associated with a second order differential operator

Let $\text{L =}\text{d}{}^2\text{dx}{}^2\text{? q(x)}$ be a Sturm-Liouville operator acting on functions defined on R. The authors have recently shown how to construct commutative associative algebras of distributions of compact support for which L is a centralizer (in the sense that $\text{L(f 1 g) = (Lf) 1 g}$ for distributions f, g of compact support) when q is locally bounded. Here, it is assumed either that q is bounded and $\text{x → (1 + ¦ x ¦) q(x)}$ is integrable, or that q is of bounded variation. A function ψ is then found such that $\text{M}$ψ={μ : μ is a measure on R and | μ |(ψ) < & infin;} becomes a Banach algebra containing the algebra of measures of compact support. The representation theory of $\text{M}$_ψ is discussed and conditions for its semisimplicity are obtained.     

Green's function and invariant density for an integro-differential operator of second order

Summary We construct the Green function for a parabolic- elliptic integro- differential operator of second order with boundary conditions given by a first order operator, both with Hölder continuous coefficients. Also, we prove the existence of a unique invariant density associated with the corresponding reflected diffusion process with jumps. This allows us to study the asymptotic behaviour of the solution for some oblique derivative problems.This research have been supported in part by Ministero P.I. « Progetto Caleolo delle Variazioni » and N.S.F. under grant No. DMS-8702236.     

On identity for eigenvalues of second order differential operator equation

In this study we investigate the spectrum and regularized trace of the second order differential operator equation with periodic boundary conditions.     

Spectral asymptotics of a nonself-adjoint elliptic differential operator with an indefinite weight function

In the present paper, we compute the leading term of the asymptotics of the angular eigenvalue distribution function of the problem Au = λω(x)u(x) in a bounded domain Ω ? R ⁿ , where A is an elliptic differential operator of order 2m with domain D(A) ? W _m ^2m (Ω). The weight function ω(x) (x ∈ Ω) is indefinite and can also take zero values on a set of positive measure.     

Schauder estimates for a degenerate second order elliptic operator on a cube

In the present article we are concerned with a class of degenerate second order differential operators LA,b defined on the cube d[0,1], with d?1. Under suitable assumptions on the coefficients A and b (among them the assumption of their Hölder regularity) we show that the operator LA,b defined on C²(d[0,1]) is closable and its closure is m-dissipative. In particular, its closure is the generator of a C₀-semigroup of contractions on C(d[0,1]) and C²(d[0,1]) is a core for it. The proof of such result is obtained by studying the solvability in Hölder spaces of functions of the elliptic problem λu(x)−LA,bu(x)=f(x), x∈d[0,1], for a sufficiently large class of functions f.