Singular Green Operators Research Articles

Singular Green operators in the edge algebra formalism

We give a new description of the class of singular Green operators introduced by Grubb in the study of parameter-dependent boundary value problems, employing the concept of operator-valued symbols introduced by Schulze in the framework of pseudodifferential operators on manifolds with edges.

Boundary singularity of Poisson and harmonic Bergman kernels

We give a complete description of the boundary behaviour of the Poisson kernel and the harmonic Bergman kernel of a bounded domain with smooth boundary, which in some sense is an analogue of the similar description for the usual Bergman kernel on a strictly pseudoconvex domain due to Fefferman. Our main tool is the Boutet de Monvel calculus of pseudodifferential boundary operators, and in fact we describe the boundary singularity of a general potential, trace or singular Green operator from that calculus.

Spectral Asymptotics for Nonsmooth Singular Green Operators

Singular Green operators G appear typically as boundary correction terms in resolvents for elliptic boundary value problems on a domain Ω ⊂ ℝ n , and more generally they appear in the calculus of pseudodifferential boundary problems. In particular, the boundary term in a Krein resolvent formula is a singular Green operator. It is well-known in smooth cases that when G is of negative order −t on a bounded domain, its eigenvalues or s-numbers have the behavior (*)s j (G) ∼ cj −t/(n−1) for j → ∞, governed by the boundary dimension n − 1. In some nonsmooth cases, upper estimates (**)s j (G) ≤ Cj −t/(n−1) are known. We show that (*) holds when G is a general selfadjoint nonnegative singular Green operator with symbol merely Hölder continuous in x. We also show (*) with t = 2 for the boundary term in the Krein resolvent formula comparing the Dirichlet and a Neumann-type problem for a strongly elliptic second-order differential operator (not necessarily selfadjoint) with coefficients in for some q > n.

Spectral asymptotics for Robin problems with a discontinuous coefficient

The spectral behavior of the difference between the resolvents of two realizations of a second-order strongly elliptic symmetric differential operator A defined by different Robin conditions \chi u=b_1\gamma_0u and \chi u=b_2\gamma_0u , can in the case where all coefficients are C^\infty be determined by use of a general result by the author in 1984 on singular Green operators. We here treat the problem for nonsmooth b_i , showing that if b_1 and b_2 are in L_\infty , the s-numbers s_j satisfy s_j j^{3/(n-1)}\le C for all j . This improves a recent result for A=-\Delta by Behrndt et al., that \sum_js_j ^p<\infty for p>(n-1)/3 , under a hypothesis of boundedness of b_i^{-1} . Moreover, we show that if b_1 and b_2 are in C^\varepsilon for some \varepsilon >0 , with jumps at a smooth hypersurface, then s_j j^{3/(n-1)}\to c for j\to \infty , with a constant defined from the principal symbol of A and b_2-b_1 . We also show that the usual principal spectral asymptotic estimate for pseudodifferential operators of negative order on a closed manifold extends to products of pseudodifferential operators of negative order interspersed with piecewise continuous functions.

Perturbation of essential spectra of exterior elliptic problems

For a second-order symmetric strongly elliptic differential operator on an exterior domain in ℝ n , it is known from the works of Birman and Solomiak that a change in the boundary condition from the Dirichlet condition to an elliptic Neumann or Robin condition leaves the essential spectrum unchanged, in such a way that the spectrum of the difference between the inverses satisfies a Weyl-type asymptotic formula. We show that one can increase, but not diminish, the essential spectrum by imposition of other Neumann-type nonelliptic boundary conditions. The results are extended to 2m-order operators, where it is shown that for any selfadjoint realization defined by an elliptic normal boundary condition (other than the Dirichlet condition), one can augment the essential spectrum at will by adding a suitable operator to the mapping from free Dirichlet data to Neumann data. We also show here an extension of the spectral asymptotics formula for the difference between inverses of elliptic problems. The proofs rely on Kreĭn-type formulae for differences between inverses, and cutoff techniques, combined with results on singular Green operators and their spectral asymptotics.

Boundedness of imaginary powers of the stokes operator in an infinite layer

In this article we prove the existence of bounded purely imaginary powers of the Stokes operator \( A_q \), which is defined on the space of solenoidal vector fields \( J_q(\Omega), 1 \) < q < \( \infty \), where \( \Omega={\hbox{\opens R}}^{n-1}\times (-1,1) \) is an infinite layer. It is a consequence of a special representation of the resolvent of the Stokes operator in terms of the Stokes operator on \( \mathbb{R}^n \), a composition of a trace and a Poisson operator – a singular Green operator – and a negligible part.

A Weakly Polyhomogeneous Calculus for Pseudodifferential Boundary Problems

In a joint work with R. Seeley, a calculus of weakly parametric pseudodifferential operators on closed manifolds was introduced and used to obtain complete asymptotic expansions of traces of resolvents and heat operators associated with the Atiyah–Patodi–Singer problem. The present paper establishes a generalization to pseudodifferential boundary operators, defining weakly polyhomogeneous singular Green operators, Poisson operators, and trace operators associated with a manifold with boundary, as well as a suitable transmission condition for pseudodifferential operators. Full composition formulas are established for the calculus, which contains the resolvents of APS-type problems. The operators in the calculus have complete asymptotic trace expansions in the parameter (when of trace class), with polynomial and logarithmic terms.

Non-local boundary conditions in Euclidean quantum gravity

Non-local boundary conditions for Euclidean quantum gravity are proposed, consisting of an integro-differential boundary operator acting on metric perturbations. In this case, the operator P on metric perturbations is of Laplace type, subject to non-local boundary conditions; in contrast, its adjoint is the sum of a Laplacian and of a singular Green operator, subject to local boundary conditions. Self-adjointness of the boundary value problem is correctly formulated by looking at Dirichlet-type and Neumann-type realizations of the operator P, following recent results in the literature. The set of non-local boundary conditions for perturbative modes of the gravitational field is written in general form on the Euclidean 4-ball. For a particular choice of the non-local boundary operator, explicit formulae for the boundary value problem are obtained in terms of a finite number of unknown functions, but subject to some consistency conditions. Among the related issues, the problem arises of whether non-local symmetries exist in Euclidean quantum gravity.

Elliptic Boundary Problems and the Boutet de Monvel Calculus in Besov and Triebel-Lizorkin Spaces.

The Boutet de Monvel calculus of pseudo-differential boundary operators is generalised to the full scales of Besov and Triebel--Lizorkin spaces (though with finite integral exponents for the latter). The continuity and Fredholm properties proved here extend those previously obtained by Franke and Grubb, and the results on range complements of surjectively elliptic Green operators improve the earlier known, even for the classical spaces with $1<p<\infty$. The symbol classes treated are the uniformly estimated ones. Some precisions are given for the general definitions of trace and singular Green operators of class 0.

A characterization of the singular green operators in boutet de monvel's calculus via wedge sobolev spaces

The singular Green operators in Boutet de Monvel's calculus are characterized by the behavior of their iterated commutators with differentiations and vector fields tangential to the boundary on wedge Sobolev spaces.

Reduction of the fundamental initial-boundary-value problems for Stokes equations to initial-boundary-value problems for parabolic systems of pseudodifferential equations

It is shown that an initial-boundary-value problem for Stokes' system, in which on the boundary one prescribes the vector field of velocities $$\vec v$$ , or the stress field, or the normal component of the velocity and the tangential stresses, reduces to an initial-boundary-value problem for a system of the form $$\vec v_t + A\vec v = \vec f$$ , where the operator A contains a nonlocal term (the so-called singular Green operator). For the solutions of these problems, coercive estimates in the spaces W2 l, l/2 and also estimates of the norm of the resolving operator in W2 r are obtained.

ASYMPTOTICS OF THE SPECTRUM OF LINEAR OPERATOR PENCILS

The problem Au = tBu is considered in a bounded Lipschitz domain, where A and B are sums of a pseudodifferential operator satisfying a transmission condition and a singular Green operator, with A elliptic. Under natural conditions the classical formula for the asymptotics of the spectrum is established, with an estimate of the remainder determined by the character of degeneration in ellipticity of the operator B.Bibliography: 18 titles.

Singular Green operators and their spectral asymptotics

Singular Green operators and their spectral asymptotics