Some compactness results by elliptic operators

This paper is concerned with the integration (of 1-forms) against the Markov stochastic process associated with a second-order elliptic differential operator in divergence form. It focuses on the limiting behavior of the integral as the process leaves a fixed point or goes to infinity. This extends previous work in the area where advantage was usually taken of the fact that the operator was self adjoint and started with the associated invariant measure. Applications are given. For example, it is a trivial consequence that the diffusion associated to a uniformly elliptic operator on a negatively curved Cartan–Hadamard manifold has an asymptotic direction (recovering and strengthening the previous arguments of Pratt, Sullivan and others). The approach can also be used to construct a Lévy area for such processes, to study the thinness of sets for the elliptic operator, and probably has wider applications.

Let \Omega \subset \mathbb{R}^d be a bounded open set with Lipschitz boundary \Gamma . It will be shown that the Jordan chains of m-sectorial second-order elliptic partial differential operators with measurable coefficients and (local or non-local) Robin boundary conditions in L_2(\Omega) can be characterized with the help of Jordan chains of the Dirichlet-to-Neumann map and the boundary operator from H^{1/2}(\Gamma) into H^{-1/2}(\Gamma) . This result extends the Birman–Schwinger principle in the framework of elliptic operators for the characterization of eigenvalues, eigenfunctions and geometric eigenspaces to the complete set of all generalized eigenfunctions and algebraic eigenspaces.

The spectrum of a selfadjoint, C' linear elliptic partial differential operator on a compact manifold contains only isolated eigenvalues, each having finite multiplicity. It is sometimes the case that these multiplicities are unbounded; this is common in problems arising in applications because of the high degree of symmetry usually present. The main theorem shows that the property of having only simple eigenvalues is generic for operators obtained by varying the zeroth order part of a given operator. The purpose of this article is to provide a proof of the following theorem. Main theorem. Let M be a compact, connected C' manifold without boundary. Let L be a sel/adjoint, C' linear elliptic differential operator on M. The set Ep e Cw(M): all eigenvalues of L + p are simple} is residual in C'(M). The term elliptic operator used in this article will always mean the operator is selfadjoint, C' and linear. The main theorem can be summarized by saying that almost all elliptic operators obtained by varying the zeroth order part of a given operator have only simple eigenvalues. For terminology, see ? 1. The main theorem was first announced in [1] for second-order operators, and the proof is the one given in [2]; it uses perturbation theory. A proof using transversality theory, applicable to operators whose eigenfunctions satisfy the strong unique continuation property, has been obtained by K. Uhlenbeck [6]. The theorem is particularly useful in obtaining genericity results about the eigenfunctions (see [1] , [2], [6]). 1. Notation; outline of proof. The manifold M and the operator L will be fixed throughout as in the theorem. For standard terminology on elliptic operators, see [3], [4]. Received by the editors June 4, 1973 and, in revised form, January 23, 1974. AMS (MOS) subject classifications (1970). Primary 35P05, 35J30; Secondary 47A55, 47B25, 58G99.

The use of diagonal scalings of the Laplacian matrix as preconditioners for matrices arising from other second-order self adjoint elliptic differential operators is considered. It is proved that if a diffusion operator with a piecewise constant but discontinuous diffusion coefficient is preconditioned by a diagonal scaling of the Laplacian, then, in the limit as the mesh size goes to zero, the optimal diagonal scaling is just the identity. If, on the other hand, the Laplacian is scaled on each side by the square root of the diagonal of the matrix corresponding to the diffusion operator, then the condition number of the preconditioned matrix grows like $O(h^{ - 2} )$, instead of $O(1)$. This is in contrast to the case in which the diffusion coefficient is smoothly varying, in which case numerical evidence suggests that the optimal diagonal scaling is approximately equal to the square root of the diagonal of the matrix.

Spaces of ultrasmooth vectors of some elliptic differential operators are described. In particular, elliptic operators with strong singular coefficients in the neighborhood of the boundary and generalized Legendre differential operators are considered.

We consider one-dimensional inhomogeneous parabolic equations with higher-order elliptic differential operators subject to periodic boundary conditions. In our main result we show that the property of continuous maximal regularity is satisfied in the setting of periodic little-H\"older spaces, provided the coefficients of the differential operator satisfy minimal regularity assumptions. We address parameter-dependent elliptic equations, deriving invertibility and resolvent bounds which lead to results on generation of analytic semigroups. We also demonstrate that the techniques and results of the paper hold for elliptic differential operators with operator-valued coefficients, in the setting of vector-valued functions.

Representations of regularized determinants of elements of one-parameter operator semigroups whose generators are second-order elliptic differential operators by Lagrangian functional integrals are obtained. Such semigroups describe solutions of inverse Kolmogorov equations for diffusion processes. For self-adjoint elliptic operators, these semigroups are often called Schrodinger semigroups, because they are obtained by means of analytic continuation from Schrodinger groups. It is also shown that the regularized determinant of the exponential of the generator (this exponential is an element of a one-parameter semigroup) coincides with the exponential of the regularized trace of the generator.

In this expository article, we consider first order elliptic differential operators acting on smooth vector bundles over compact manifolds, and certain invariants derived from the analysis of these operators, namely the eta invariant} and the equivariant index. Many researchers have previously considered these invariants before. What makes this work different is that we are evaluating integer-valued indices corresponding to multiplicities of group representations, and our eta invariant is a number dependent on the entire group at once. Moreover, the techniques of proof and formulas obtained are new and depend on equivariant heat asymptotics that may involve logarithmic terms. For simplicity, we consider only elliptic differential operators, even though the proofs outlined apply to transversally elliptic operators. In every case, we outline the well-known proofs and theorems without Lie group actions first and then show how these same ideas can be applied in the equivariant cases with appropriate modifications. A more detailed and expanded article that applies to transversally elliptic operators will appear in due time.

In this paper we obtain pointwise two-sided estimates for the integral kernel of the semigroup associated with second-order elliptic differential operators −∇⋅(a∇)+b 1⋅∇+∇⋅b 2+V with real measurable (singular) coefficients, on an open set Ω⊂R N . The assumptions we impose on the lower-order terms allow for the case when the semigroup exists on L p (Ω) for p only from an interval in [1,∞), neither enjoys a standard Gaussian estimate nor is ultracontractive in the scale L p (Ω). We show however that the semigroup is ultracontractive in the scale of weighted spaces L p (Ω,ϕ2 dx) with a suitable weight ϕ and derive an upper and lower bound on its integral kernel.

Chebyshev expansion methods for the solution of the extended graetz problem

We consider the problem of homogenization for non-self-adjoint second-order elliptic differential operators $\mathcal{A}^{\varepsilon}$ of divergence form on $L_{2}(\mathbb{R}^{d_{1}}\times\mathbb{T}^{d_{2}})$, where $d_{1}$ is positive and $d_{2}$ is non-negative. The coefficients of the operator $\mathcal{A}^{\varepsilon}$ are periodic in the first variable with period $\varepsilon$ and smooth in a certain sense in the second. We show that, as $\varepsilon$ gets small, $(\mathcal{A}^{\varepsilon}-\mu)^{-1}$ and $\nabla_{x_{2}}(\mathcal{A}^{\varepsilon}-\mu)^{-1}$ for an appropriate $\mu$ converge in the operator norm to, respectively, $(\mathcal{A}^{0}-\mu)^{-1}$ and $\nabla_{x_{2}}(\mathcal{A}^{0}-\mu)^{-1}$, where $\mathcal{A}^{0}$ is an operator whose coefficients depend only on $x_{2}$. We also obtain an approximation for $\nabla_{x_{1}}(\mathcal{A}^{\varepsilon}-\mu)^{-1}$ and find the next term in the approximation for $(\mathcal{A}^{\varepsilon}-\mu)^{-1}$. Estimates for the rates of convergence ...

Analytic K-theory on manifolds with corners

We describe a homogenization model of an elastic membrane reinforced by the inclusion of a fractal string. We follow a variational approach consisting in proving the convergence of certain energy functionals. This leads to the spectral convergence of a sequence of weighted second-order elliptic partial differential operators to a singular elliptic operator with a fractal term.

In this survey some singular homogenization results are described. This approach leads to the spectral convergence of a sequence of weighted second-order elliptic partial differential operators to a singular elliptic operator with a fractal term.

Suppose that L is a second-order self-adjoint elliptic partial differential operator on a bounded domain Ω ⊂ Rn, n ≥ 2, and a, b ∈ L∞(Ω). If the equation Lu = au+ − bu− + λu (where λ ∈ R and u±(x) = max{±u(x), 0}) has a non-trivial solution u, then λ is said to be a half-eigenvalue of (L; a, b). In this paper, we obtain some general properties of the half-eigenvalues of (L; a, b) and also show that, generically, the half-eigenvalues are ‘simple’.We also consider the semilinear problem where f : Ω × R → R is a Carathéodory function such that, for a.e. x ∈ Ω, and we relate the solvability properties of this problem to the location of the half-eigenvalues of (L; a, b).