Self-adjoint Elliptic Operators Research Articles

Approximations of delocalized eta invariants by their finite analogues

For a given self-adjoint first order elliptic differential operator on a closed smooth manifold, we prove a list of results on when the delocalized eta invariant associated to a regular covering space can be approximated by the delocalized eta invariants associated to finite-sheeted covering spaces. One of our main results is the following. Suppose M is a closed smooth spin manifold and \widetilde{M} is a \Gamma -regular covering space of M . Let \langle \alpha \rangle be the conjugacy class of a non-identity element \alpha\in \Gamma . Suppose \{\Gamma_i\} is a sequence of finite-index normal subgroups of \Gamma that distinguishes \langle \alpha \rangle . Let \pi_{\Gamma_i} be the quotient map from \Gamma to \Gamma/\Gamma_i and \langle \pi_{\Gamma_i}(\alpha) \rangle the conjugacy class of \pi_{\Gamma_i}(\alpha) in \Gamma/\Gamma_i . If the scalar curvature on M is everywhere bounded below by a sufficiently large positive number, then the delocalized eta invariant for the Dirac operator of \widetilde{M} at the conjugacy class \langle \alpha \rangle is equal to the limit of the delocalized eta invariants for the Dirac operators of M_{\Gamma_i} at the conjugacy class \langle \pi_{\Gamma_i}(\alpha) \rangle , where M_{\Gamma_i}= \widetilde{M}/\Gamma_i is the finite-sheeted covering space of M determined by \Gamma_i . In another main result of the paper, we prove that the limit of the delocalized eta invariants for the Dirac operators of M_{\Gamma_i} at the conjugacy class \langle \pi_{\Gamma_i}(\alpha) \rangle converges, under the assumption that the rational maximal Baum–Connes conjecture holds for \Gamma .

Self-adjoint local boundary problems on compact surfaces. II. Family index

The paper presents a first step towards a family index theorem for classical self-adjoint boundary value problems.We address here the simplest non-trivial case of manifolds with boundary, namely the case of two-dimensional manifolds. The first result of the paper is an index theorem for families of first order self-adjoint elliptic differential operators with local boundary conditions, parametrized by points of a compact topological space X . We compute the K^1(X) -valued index in terms of the topological data over the boundary. The second result is the universality of the index: we show that the index is a universal additive homotopy invariant for such families if the vanishing on families of invertible operators is assumed.

Shape Sensitivity of Eigenvalue Functionals for Scalar Problems: Computing the Semi-derivative of a Minimum

This paper is devoted to the computation of certain directional semi-derivatives of eigenvalue functionals of self-adjoint elliptic operators involving a variety of boundary conditions. A uniform treatment of these problems is possible by considering them as a problem of calculating the semi-derivative of a minimum with respect to a parameter. The applicability of this approach, which can be traced back to the works of Danskin [8, 9] and Zolésio [28], to the treatment of eigenvalue problems (where the full shape derivative may not exist, due to multiplicity issues), has been illustrated by Zolésio in [29] (see also [10, Chap. 10] and included references). Despite this, some of the recent literature (see, for example, [1] or [7]) on the shape sensitivity of eigenvalue problems still continue to employ methods such as the material derivative method or Lagrangian methods which seem less adapted to this class of problems. The Delfour–Zolésio approach does not seem to be fully exploited in the existing literature: we aim to recall the importance and the simplicity of the ideas from [8, 28], by applying it to the analysis of the shape sensitivity for eigenvalue functionals for a class of elliptic operators in the scalar setting (Laplacian or diffusion in heterogeneous media), thus recovering known results in the case of Dirichlet or Neumann boundary conditions and obtaining new results in the case of Steklov or Wentzell boundary conditions.

Analytic Resolving Families for Equations with Distributed Riemann–Liouville Derivatives

Some new necessary and sufficient conditions for the existence of analytic resolving families of operators to the linear equation with a distributed Riemann–Liouville derivative in a Banach space are established. We study the unique solvability of a natural initial value problem with distributed fractional derivatives in the initial conditions to corresponding inhomogeneous equations. These abstract results are applied to a class of initial boundary value problems for equations with distributed derivatives in time and polynomials with respect to a self-adjoint elliptic differential operator in spatial variables.

On the Unique Solvability of Incomplete Cauchy Type Problems for a Class of Multi-Term Equations with the Riemann–Liouville Derivatives

Incomplete Cauchy-type problems are considered for linear multi-term equations solved with respect to the highest derivative in Banach spaces with fractional Riemann–Liouville derivatives and with linear closed operators at them. Some new existence and uniqueness theorems for solutions are presented explicitly and the analyticity of the solutions of the homogeneous equations are also shown. The asymmetry of the Cauchy-type problem under study is expressed in the presence of a so-called defect, which shows the number of lower-order initial conditions that should not be set when setting the problem. As applications, our abstract results are used in the study of a class of initial-boundary value problems for multi-term equations with Riemann–Liouville derivatives in time and with polynomials of a self-adjoint elliptic differential operator with respect to spatial variables.

Spectra of Elliptic Operators on Quantum Graphs with Small Edges

We consider a general second order self-adjoint elliptic operator on an arbitrary metric graph, to which a small graph is glued. This small graph is obtained via rescaling a given fixed graph γ by a small positive parameter ε. The coefficients in the differential expression are varying, and they, as well as the matrices in the boundary conditions, can also depend on ε and we assume that this dependence is analytic. We introduce a special operator on a certain extension of the graph γ and assume that this operator has no embedded eigenvalues at the threshold of its essential spectrum. It is known that under such assumption the perturbed operator converges to a certain limiting operator. Our main results establish the convergence of the spectrum of the perturbed operator to that of the limiting operator. The convergence of the spectral projectors is proved as well. We show that the eigenvalues of the perturbed operator converging to limiting discrete eigenvalues are analytic in ε and the same is true for the associated perturbed eigenfunctions. We provide an effective recurrent algorithm for determining all coefficients in the Taylor series for the perturbed eigenvalues and eigenfunctions.

Initial Value Problems of Linear Equations with the Dzhrbashyan–Nersesyan Derivative in Banach Spaces

Among the many different definitions of the fractional derivative, the Riemann–Liouville and Gerasimov–Caputo derivatives are most commonly used. In this paper, we consider the equations with the Dzhrbashyan–Nersesyan fractional derivative, which generalizes the Riemann–Liouville and the Gerasimov–Caputo derivatives; it is transformed into such derivatives for two sets of parameters that are, in a certain sense, symmetric. The issues of the unique solvability of initial value problems for some classes of linear inhomogeneous equations of general form with the fractional Dzhrbashyan–Nersesyan derivative in Banach spaces are investigated. An inhomogeneous equation containing a bounded operator at the fractional derivative is considered, and the solution is presented using the Mittag–Leffler functions. The result obtained made it possible to study the initial value problems for a linear inhomogeneous equation with a degenerate operator at the fractional Dzhrbashyan–Nersesyan derivative in the case of relative p-boundedness of the operator pair from the equation. Abstract results were used to study a class of initial boundary value problems for equations with the time-fractional Dzhrbashyan–Nersesyan derivative and with polynomials in a self-adjoint elliptic differential operator with respect to spatial variables.

On approximate controllability of a class of degenerate fractional order distributed systems

The approximate controllability issues for a class of control systems, whose dynamics is described by an equation in a Banach space with a linear degenerate operator at the Riemann — Liouville fractional derivative, is investigated. Under the condition of p-boundedness of the pair of operators in the equation the control system is reduced to subsystems on two mutually complement subspaces. It was shown that the approximate controllability of the whole system is equivalent to the approximate controllability of the two subsystems. Criteria of the approximate controllability for the system and two subsystems are obtained. Analogous results are got on the approximate controllability for free time and for systems of the same form with a finite-dimensional input. The obtained criteria were applied to the investigation of the approximate controllability for a distributed system with polynomials of a differential with respect to the spatial variables self-adjoint elliptic operator and for a system of the Scott-Blair — Oskolkov type.

Asymptotic behavior of repeated eigenvalues of perturbed self-adjoint elliptic operators

A variational characterization for the shift of eigenvalues caused by a general type of perturbation is derived for second order self-adjoint elliptic differential operators. This result allows the direct extension of asymptotic formulae from simple eigenvalues to repeated ones. Some examples of particular interest are presented theoretically and numerically for the Laplacian operator for the following domain perturbations: excision of a small hole, local change of conductivity, small boundary deformation.

Линейные обратные задачи для уравнений с несколькими производными Римана – Лиувилля

The issues of well-posedness of linear inverse coefficient problems for multi-term equations in Banach spaces with fractional Riemann – Liouville derivatives and with bounded operators at them are considered. Well-posedness criteria are obtained both for the equation resolved with respect to the highest fractional derivative, and in the case of a degenerate operator at the highest derivative in the equation. Two essentially different cases are investigated in the degenerate problem: when the fractional part of the order of the second-oldest derivative is equal to or different from the fractional part of the order of the highest fractional derivative. Abstract results are applied in the study of inverse problems for partial differential equations with polynomials from a self-adjoint elliptic differential operator with respect to spatial variables and with Riemann – Liouville derivatives in time.

Weyl Law on Asymptotically Euclidean Manifolds

We study the asymptotic behaviour of the eigenvalue counting function for self-adjoint elliptic linear operators defined through classical weighted symbols of order (1, 1), on an asymptotically Euclidean manifold. We first prove a two-term Weyl formula, improving previously known remainder estimates. Subsequently, we show that under a geometric assumption on the Hamiltonian flow at infinity, there is a refined Weyl asymptotics with three terms. The proof of the theorem uses a careful analysis of the flow behaviour in the corner component of the boundary of the double compactification of the cotangent bundle. Finally, we illustrate the results by analysing the operator Q=(1+|x|^2)(1-varDelta ) on mathbb {R}^d.

Neumann fractional diffusion problems: BURA solution methods and algorithms

Let us consider the non-local problem −Lαu=f, α∈(0,1), L is a second order self-adjoint elliptic operator in Ω⊂Rd with Neumann boundary conditions on ∂Ω. The problem is discretized by finite difference or finite element method, thus obtaining the linear system Aαu=f, A is sparse symmetric and positive semidefinite matrix. The proposed method is based on best uniform rational approximations (BURA) of degree k, rα,k, of the scalar function tα, t∈[0,1]. Then, the approximate solution ur of the fractional power linear system is defined as u≈ur=λ2−αrα,k(λ2A†)f, where λ2 is the first positive eigenvalue of A, and A† stands for the Moore–Penrose pseudo inverse of A. The BURA method reduces the non-local problem to solution of k linear systems with matrices A+diI, di>0, i=1,…,k. An exponential convergence rate with respect to k is proven. The error estimates are robust with respect to the condition number of A in the subspace orthogonal to the constant vectors. The algorithm has almost optimal computational complexity assuming that optimal iterative solvers are applied to the auxiliary sparse linear systems. The first group of numerical tests illustrates in detail the obtained theoretical results. Finite difference discretization of a model 2D problem is used for this purpose. The second part of numerical tests demonstrates the applicability of BURA methods for 3D problems in domains with general geometry. Linear finite elements on unstructured tetrahedral meshes with local mesh refinement are used in the presented large-scale experiments, confirming also the almost optimal computational complexity.

Guaranteed a posteriori bounds for eigenvalues and eigenvectors: Multiplicities and clusters

This paper presents a posteriori error estimates for conforming numerical approximations of eigenvalue clusters of second-order self-adjoint elliptic linear operators with compact resolvent. Given a cluster of eigenvalues, we estimate the error in the sum of the eigenvalues, as well as the error in the eigenvectors represented through the density matrix, i.e., the orthogonal projector on the associated eigenspace. This allows us to deal with degenerate (multiple) eigenvalues within the framework. All the bounds are valid under the only assumption that the cluster is separated from the surrounding smaller and larger eigenvalues; we show how this assumption can be numerically checked. Our bounds are guaranteed and converge with the same speed as the exact errors. They can be turned into fully computable bounds as soon as an estimate on the dual norm of the residual is available, which is presented in two particular cases: the Laplace eigenvalue problem discretized with conforming finite elements, and a Schrödinger operator with periodic boundary conditions of the form − Δ + V -\Delta + V discretized with planewaves. For these two cases, numerical illustrations are provided on a set of test problems.

Homogenization of periodic parabolic systems in the $L_2(\mathbb {R}^d)$-norm with the corrector taken into account

In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, consider a self-adjoint matrix second order elliptic differential operator $\mathcal{B}_\varepsilon$, $0<\varepsilon \leqslant 1$. The principal part of the operator is given in a factorised form, the operator contains first and zero order terms. The operator $\mathcal{B}_\varepsilon$ is positive definite, its coefficients are periodic and depend on $\mathbf{x}/\varepsilon$. We study the behaviour in the small period limit of the operator exponential $e^{-\mathcal{B}_\varepsilon t}$, $t\geqslant 0$. The approximation in the $(L_2\rightarrow L_2)$-operator norm with error estimate of order $O(\varepsilon ^2)$ is obtained. The corrector is taken into account in this approximation. The results are applied to homogenization of the solutions for the Cauchy problem for parabolic systems.

Elliptic Operators in Multidimensional Cylinders with Frequently Alternating Boundary Conditions Along a Given Curve

We consider a selfadjoint elliptic operator in an infinite multidimensional cylinder with the Dirichlet boundary condition which is replaced by the Robin condition on small sets located along a given line on the boundary. The shape and distribution of these sets are arbitrary. The characteristic linear size of these sets is a small parameter of the problem. It is shown that the resolvent of such an operator converges to the resolvent of the homogenized operator, and an estimate for the convergence rate is obtained. The homogenized operator is the same operator, but without alternating boundary conditions. The difference of resolvents is estimated in the norm of bounded operators acting from L2 to $$ {W}_2^1. $$ If the small sets are periodically distributed and have the same shape and the Robin condition is replaced by the Neumann condition, we derive two-sided asymptotic estimates for the lower band functions, which provides the possibility to estimate from below the length of the first band of the spectrum.

Self-Adjoint Local Boundary Problems on Compact Surfaces. I. Spectral Flow

The paper deals with first order self-adjoint elliptic differential operators on a smooth compact oriented surface with non-empty boundary. We consider such operators with self-adjoint local boundary conditions. The paper is focused on paths in the space of such operators connecting two operators conjugated by a unitary automorphism. The first result is the computation of the spectral flow for such paths in terms of the topological data over the boundary. The second result is the universality of the spectral flow: we show that the spectral flow is a universal additive invariant for such paths, if the vanishing on paths of invertible operators is required. In the next paper of the series we generalize these results to families of such operators parametrized by points of an arbitrary compact space instead of an interval. The integer-valued spectral flow is replaced then by the family index taking values in the $K^1$-group of the base space.

The index of generalised Dirac–Schrödinger operators

We study the relation between spectral flow and index theory within the framework of (unbounded) K K-theory. In particular, we consider a generalised notion of ‘Dirac–Schrödinger operators’, consisting of a self-adjoint elliptic first-order differential operator \mathcal D with a skew-adjoint ‘potential’ given by a (suitable) family of unbounded operators on an auxiliary Hilbert module. We show that such Dirac–Schrödinger operators are Fredholm, and we prove a relative index theorem for these operators (which allows cutting and pasting of the underlying manifolds). Furthermore, we show that the index of a Dirac–Schrödinger operator represents the pairing (Kasparov product) of the K -theory class of the potential with the K -homology class of \mathcal D . We prove this result without assuming that the potential is differentiable; instead, we assume that the ‘variation’ of the potential is sufficiently small near infinity. In the special case of the real line, we recover the well-known equality of the index with the spectral flow of the potential.

Error estimates of the quadrature finite element method with biquadratic finite elements for elliptic eigenvalue problems in the square domain

A positive definite eigenvalue problem for second-order self-adjoint elliptic differential operator defined on a square domain in the plane with homogeneous Dirichlet boundary condition is considered. This problem has a nondecreasing sequence of positive eigenvalues of finite multiplicity with a limit point at infinity. To the sequence of eigenvalues, there corresponds an orthonormal system of eigenfunctions. The original differential eigenvalue problem is approximated by the finite element method with numerical integration and Lagrange biquadratic finite elements. Error estimates for approximate eigenvalues and eigenfunctions are established.

The Singular Perturbation Problem for a Class of Generalized Logistic Equations Under Non-classical Mixed Boundary Conditions

Abstract This paper studies a singular perturbation result for a class of generalized diffusive logistic equations, d ⁢ ℒ ⁢ u = u ⁢ h ⁢ ( u , x ) {d\mathcal{L}u=uh(u,x)} , under non-classical mixed boundary conditions, ℬ ⁢ u = 0 {\mathcal{B}u=0} on ∂ ⁡ Ω {\partial\Omega} . Most of the precursors of this result dealt with Dirichlet boundary conditions and self-adjoint second order elliptic operators. To overcome the new technical difficulties originated by the generality of the new setting, we have characterized the regularity of ∂ ⁡ Ω {\partial\Omega} through the regularity of the associated conormal projections and conormal distances. This seems to be a new result of a huge relevance on its own. It actually complements some classical findings of Serrin, [39], Gilbarg and Trudinger, [21], Krantz and Parks, [27], Foote, [18] and Li and Nirenberg [28] concerning the regularity of the inner distance function to the boundary.

Spectral analysis of discrete elliptic operators and applications in control theory

In this paper we propose an analysis of discrete spectral properties for a finite difference discretization of quite general 1D second-order self-adjoint elliptic operators. We particularly investigate some (uniform in the discretization parameter) qualitative behavior of eigenfunctions and eigenvalues. With those estimates we manage to obtain new results for the construction of bounded families of controls for semi-discrete parabolic PDEs, in particular for boundary controls of a coupled parabolic system with fewer controls than equations.