Edge Operators in Geometry

Analytic K-theory on manifolds with corners

Examples of edge operators include Laplacians on asymptotically flat and asymptotically hyperbolic manifolds. Edge operators also arise in boundary problems around higher condimension boundaries. This paper is concerned with the analysis of general elliptic edge operators with constant indicide roots. We determine when such an operator has a distributional asymptotic expansion. Conditions are given to guarantee that the coefficients of this expansion are smooth. In Part I of this paper we only study the case when the operator is semi-Fredholm. Part II will examine edge operators with infinite dimensional kernel and cokernel, as well as develop the theory of Poisson edge operators.

Partial differential operators, and in particular elliptic operators, on a Riemannian manifold provide a rich source of problems of non-commutative analysis. In the simplest situation the manifold is a Lie group and one can examine elliptic operators constructed from a vector space basis of the Lie algebra or subelliptic operators constructed from a Lie algebraic basis. In the latter case the choice of algebraic basis affects the characteristic geometric features of the subelliptic operators and their description involves an interesting interplay between analysis and geometry.

definitions are recalled in the section of preliminaries in this paper. In this paper we compute the ring of differential operators for some noncommutative rings, namely the Axumaya algebras and the Heisenberg algebras. The initial interest was to compute these rings for matrix algebras and the Weyl algebras, which were easily generalized. We consider Azumaya algebras over a noetherian ring. We conclude that the ring of differential operators are generated as modules by the ring of differential operators on their centre and homomorphisms given by multiplication by elements of the bigger ring (called inner homomorphisms). That is, if R is the centre of A (where A is an Azumaya algebra over R), we show that Dk(R) can be embedded into Dk(A) and that Dk(A) = (A N R A o ).Dk(R).(A N R A o ), that is, Dk(A) is gener

This study discusses generalized Rayleigh quotient and high efficiency finite element discretization schemes. Some results are as follows: 1) Rayleigh quotient accelerate technique is extended to nonselfadjoint problems. Generalized Rayleigh quotients of operator form and weak form are defined and the basic relationship between approximate eigenfunction and its generalized Rayleigh quotient is established. 2) New error estimates are obtained by replacing the ascent of exact eigenvalue with the ascent of finite element approximate eigenvalue. 3) Based on the work of Xu Jinchao and Zhou Aihui, finite element two-grid discretization schemes are established to solve nonselfadjoint elliptic differential operator eigenvalue problems and these schemes are used in both conforming finite element and nonconforming finite element. Besides, the efficiency of the schemes is proved by both theoretical analysis and numerical experiments. 4) Iterated Galerkin method, interpolated correction method and gradient recovery for selfadjoint elliptic differential operator eigenvalue problems are extended to nonselfadjoint elliptic differential operator eigenvalue problems.

Let $A$ be a graded $C^{\ast}$-algebra. We characterize Kasparov's $K$-theory group $\hat{K}_{0}(A)$ in terms of graded $\ast$-homomorphisms by proving a general converse to the functional calculus theorem for self-adjoint regular operators on graded Hilbert modules. An application to the index theory of elliptic differential operators on smooth closed manifolds and asymptotic morphisms is discussed.

Consider the fractional powers and of the Dirichlet and Neumann realizations of a second‐order strongly elliptic differential operator A on a smooth bounded subset Ω of . Recalling the results on complex powers and complex interpolation of domains of elliptic boundary value problems by Seeley in the 1970's, we demonstrate how they imply regularity properties in full scales of ‐Sobolev spaces and Hölder spaces, for the solutions of the associated equations. Extensions to nonsmooth situations for low values of s are derived by use of recent results on ‐calculus. We also include an overview of the various Dirichlet‐ and Neumann‐type boundary problems associated with the fractional Laplacian.

We study the solutions of equations of type f ( D, α)u = v, where f ( D, α) is a p-adic pseudodifferential operator.If v is a Bruhat-Schwartz function, there exists a distribution E α , a fundamental solution, such that u = E α * v is a solution.However, it is unknown to which function space E α * v belongs.We show that if f ( D, α) is an elliptic operator, then u = E α * v belongs to a certain Sobolev space, and we give conditions for the continuity and uniqueness of u.By modifying the Sobolev norm, we establish that f ( D, α) gives an isomorphism between certain Sobolev spaces.

Parameter-dependent variational–hemivariational inequalities and an unstable degenerate elliptic free boundary problem

In the study of elliptic differential operators of second order, we found that the automorphism pseudogroups are of finite type ([4]). However this fact takes a complete change in elliptic differential operators of first order.So as to make the objects which can be dealt with clear, we introduce the concept of pseudoellipticity of first order differential operators (Definition 1.1), which is naturally satisfied by first order elliptic differential operators.

We consider second order uniformly elliptic operators of divergence form in $\R^{d+1}$ whose coefficients are independent of one variable. Under the Lipschitz condition on the coefficients we characterize the domain of the Poisson operators and the Dirichlet-Neumann maps in the Sobolev space $H^s(\R^d)$ for each $s\in [0,1]$. Moreover, we also show a factorization formula for the elliptic operator in terms of the Poisson operator.

This paper, somewhat delayed in the series “Fundamental Directions,” is devoted to general linear elliptic boundary problems on a smooth compact manifold with boundary. The paper is intended for a large circle of readers. We hope the paper will be useful to many mathematicians with diverse scientific interests. The main features of the theory we will discuss were formed in the 60’s, beginning with the proof of the equivalence of the ellipticity conditions and the Fredholm property of the corresponding operator in the simplest Sobolev spaces. This was done on the basis of achievements of many mathematicians during the preceding decades. This basis was very extensive, but the results were incomplete. The elaboration of the general theory was stimulated by investigations of the index problem (see e.g. (Palais 1965) and (Fedosov 1990)) and went on under a strong influence of the microlocal analysis, beginning with the appearance of the calculus of pseudodifferential operators. During the last decades, new variants of the general theory appeared, and the old variants were enriched by new results.KeywordsBoundary ProblemElliptic OperatorPseudodifferential OperatorLipschitz DomainElliptic BoundaryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The study of spectral properties of natural geometric elliptic partial differential operators acting on smooth sections of vector bundles over Riemannian manifolds is a central theme in global analysis, differential geometry and mathematical physics. Instead of studying the spectrum of a differential operator [Formula: see text] directly one usually studies its spectral functions, that is, spectral traces of some functions of the operator, such as the spectral zeta function [Formula: see text] and the heat trace [Formula: see text]. The kernel [Formula: see text] of the heat semigroup [Formula: see text], called the heat kernel, plays a major role in quantum field theory and quantum gravity, index theorems, non-commutative geometry, integrable systems and financial mathematics. We review some recent progress in the study of spectral asymptotics. We study more general spectral functions, such as [Formula: see text], that we call quantum heat traces. Also, we define new invariants of differential operators that depend not only on the eigenvalues but also on the eigenfunctions, and, therefore, contain much more information about the geometry of the manifold. Furthermore, we study some new invariants, such as [Formula: see text], that contain relative spectral information of two differential operators. Finally, we show how the convolution of the semigroups of two different operators can be computed by using purely algebraic methods.

In [Spectral asymmetry and Riemannian geometry. III, Math. Proc. Cambridge Philos. Soc. 79 (1976) 71–99] Atiyah, Patodi and Singer introduced spectral flow for elliptic operators on odd-dimensional compact manifolds. They argued that it could be computed from the Fredholm index of an elliptic operator on a manifold of one higher dimension. A general proof of this fact was produced by Robbin–Salamon [The spectral flow and the Maslov index, Bull. London Math. Soc. 27(1) (1995) 1–33, MR 1331677]. In [F. Gesztesy, Y. Latushkin, K. Makarov, F. Sukochev and Y. Tomilov, The index formula and the spectral shift function for relatively trace class perturbations, Adv. Math. 227(1) (2011) 319–420, MR 2782197], a start was made on extending these ideas to operators with some essential spectrum as occurs on non-compact manifolds. The new ingredient introduced there was to exploit scattering theory following the fundamental paper [A. Pushnitski, The spectral flow, the Fredholm index, and the spectral shift function, in Spectral Theory of Differential Operators, American Mathematical Society Translations: Series 2, Vol. 225 (American Mathematical Society, Providence, RI, 2008), pp. 141–155, MR 2509781]. These results do not apply to differential operators directly, only to pseudo-differential operators on manifolds, due to the restrictive assumption that spectral flow is considered between an operator and its perturbation by a relatively trace-class operator. In this paper, we extend the main results of these earlier papers to spectral flow between an operator and a perturbation satisfying a higher pth Schatten class condition for [Formula: see text], thus allowing differential operators on manifolds of any dimension [Formula: see text]. In fact our main result does not assume any ellipticity or Fredholm properties at all and proves an operator theoretic trace formula motivated by [M.-T. Benameur, A. Carey, J. Phillips, A. Rennie, F. Sukochev and K. Wojciechowski, An analytic approach to spectral flow in von Neumann algebras, in Analysis, Geometry and Topology of Elliptic Operators (World Scientific Publisher, Hackensack, NJ, 2006), pp. 297–352, MR 2246773; A. Carey, H. Grosse and J. Kaad, On a spectral flow formula for the homological index, Adv. Math. 289 (2016) 1106–1156, MR 3439708]. We illustrate our results using Dirac type operators on [Formula: see text] for arbitrary [Formula: see text] (see Sec. 8). In this setting Theorem 6.4 substantially extends Theorem 3.5 of [A. Carey, F. Gesztesy, H. Grosse, G. Levitina, D. Potapov, F. Sukochev and D. Zanin, Trace formulas for a class of non-Fredholm operators: a review, Rev. Math. Phys. 28(10) (2016) 1630002, MR 3572626], where the case d = 1 was treated.

Convergence of iterations of special Green integrals for overdetermined elliptic linear partial differential operatorsPof order p ≥ 1 is proved. Using this result we obtain necessary and sufficient conditions for the solvability of the equationPu=fin the Sobolev space Wp,2(D) and, as a corollary, necessary and sufficient conditions for the vanishing of the first cohomology group of elliptic differential complexes. Also a criterion for the solvability of aP‐Neumann problem for elliptic differential operators is proved.