Second-order Elliptic Differential Operators Research Articles

A Convergent Adaptive Method for Elliptic Eigenvalue Problems

We prove the convergence of an adaptive linear finite element method for computing eigenvalues and eigenfunctions of second-order symmetric elliptic partial differential operators. The weak form is assumed to yield a bilinear form which is bounded and coercive in $H^1$. Each step of the adaptive procedure refines elements in which a standard a posteriori error estimator is large and also refines elements in which the computed eigenfunction has high oscillation. The error analysis extends the theory of convergence of adaptive methods for linear elliptic source problems to elliptic eigenvalue problems, and in particular deals with various complications which arise essentially from the nonlinearity of the eigenvalue problem. Because of this nonlinearity, the convergence result holds under the assumption that the initial finite element mesh is sufficiently fine.

Krein's resolvent formula for self-adjoint extensions of symmetric second-order elliptic differential operators

Given a symmetric, semi-bounded, second-order elliptic differential operator A on a bounded domain with C1,1 boundary, we provide a Kreĭn-type formula for the resolvent difference between its Friedrichs extension and an arbitrary self-adjoint one.

Fractal Reinforcement of Elastic Membranes

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.

Intrinsic ultracontractivity of nonsymmetric diffusions with measure-valued drifts and potentials

Recently, in [Preprint (2006)], we extended the concept of intrinsic ultracontractivity to nonsymmetric semigroups. In this paper, we study the intrinsic ultracontractivity of nonsymmetric diffusions with measure-valued drifts and measure-valued potentials in bounded domains. Our process Y is a diffusion process whose generator can be formally written as L + μ ∇ ¯ ν with Dirichlet boundary conditions, where L is a uniformly elliptic second-order differential operator and μ = (μ 1 ,... μ d ) is such that each component μ i , i = 1,..., d, is a signed measure belonging to the Kato class K d , and v is a (nonnegative) measure belonging to the Kato class K d,2 . We show that scale-invariant parabolic and elliptic Harnack inequalities are valid for Y. In this paper, we prove the parabolic boundary Harnack principle and the intrinsic ultracontractivity for the killed diffusion Y D with measure-valued drift and potential when D is one of the following types of bounded domains: twisted Holder domains of order a e (1/3, 1], uniformly Holder domains of order a ∈ (0, 2) and domains which can be locally represented as the region above the graph of a function. This extends the results in [J. Funct. Anal. 100 (1991) 181-206] and [Probab. Theory Related Fields 91 (1992) 405-443]. As a consequence of the intrinsic ultracontractivity, we get that the supremum of the expected conditional lifetimes of Y D is finite.

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.

Continuous selections of Borel measures, positive operators and degenerate evolution problems

In this paper we continue the study of a sequence of positive linear operators which we have introduced in [9] and which are associated with a continuous selection of Borel measures on the unit interval. We show that the iterates of these operators converge to a Markov semigroup whose generator is a degenerate second-order elliptic differential operator on the unit interval. Some qualitative properties of the semigroup, or equivalently, of the solutions of the corresponding degenerate evolution problems, are also investigated.

Why Finite Element Discretizations Can Be Factored by Triangular Hierarchical Matrices

Although the asymptotic complexity of direct methods for the solution of large sparse finite element systems arising from second-order elliptic partial differential operators is far from being optimal, these methods are often preferred over modern iterative methods. This is mainly due to their robustness. In this article it is shown that an approximate $LU$ decomposition exists which can be computed in the algebra of hierarchical matrices with almost linear complexity and with the same robustness as the classical $LU$ decomposition. Low-precision approximations may be used for preconditioning iterative solvers. As a byproduct we prove that Schur complements of stiffness matrices can be approximated with almost linear complexity.

Approximation of solutions to non-linear integrodifferential parabolic equations in $L^p$-spaces

This paper is concerned with non-linear parabolic integrodifferential equations arising in continuum mechanics, phase-field models, and elsewhere where memory terms are important. More precisely, we will consider a non-linear initial-value problem depending on a small parameter and related to a uniformly elliptic second-order differential operator $A$. The integrodifferential character of our problem is expressed by a convolution term multiplying $A$, the scalar kernel approximating a delta-type function. Consequently, the limit problem reduces to a nonlinear parabolic differential problem involving a multiple of operator $A$. Two different non-linearities, which are locally Lipschitz-continuous in suitable metrics, are considered. The basic aim of the paper consists in determining the rate of convergence of the approximating solutions to the exact one. The results proved in the linear case (cf. [10]) are used here as starting points to solve our problems.

Approximation of exact controllability problem involving parabolic differential equations

In this paper we examine computation of optimal control u* of the exact controllability problem (referred to as the constraint problem) governed by the following type of linear parabolic differential equations: (∂y/∂t) + Ay = u in Q y = 0 on ∑ y(0) = y0 on Ω where A is the second-order elliptic differential operator, Ω is a bounded domain in ℝk with smooth boundary ∂Ω, Q = (0, T) × Ω, ∑ = (0, T) × ∂Ω and T > 0. This is achieved by approximating u* through a sequence {un} of controls corresponding to unconstrained problems involving a penalty function arising from the controllability constraint.

DIRAC OPERATOR IN MATRIX GEOMETRY

We review the construction of the Dirac operator and its properties in Riemannian geometry, and show how the asymptotic expansion of the trace of the heat kernel determines the spectral invariants of the Dirac operator and its index. We also point out that the Einstein–Hilbert functional can be obtained as a linear combination of the first two spectral invariants of the Dirac operator. Next, we report on our previous attempts to generalize the notion of the Dirac operator to the case of Matrix Geometry, where, instead of a Riemannian metric there is a matrix valued self-adjoint symmetric two-tensor that plays a role of a "non-commutative" metric. We construct invariant first-order and second-order self-adjoint elliptic partial differential operators, which can be called "non-commutative" Dirac operators and non-commutative Laplace operators. We construct the corresponding heat kernel for the non-commutative Laplace type operator and compute its first two spectral invariants. A linear combination of these two spectral invariants gives a functional that can be considered as a non-commutative generalization of the Einstein–Hilbert action.

Generalized Fourier transform on an arbitrary triangular domain

In this paper, we construct generalized Fourier transform on an arbitrary triangular domain via barycentric coordinates and PDE approach. We start with a second-order elliptic differential operator for an arbitrary triangle which has the so-called generalized sine (TSin) and generalized cosine (TCos) systems as eigenfunctions. The orthogonality and completeness of the systems are then proved. Some essential convergence properties of the generalized Fourier series are discussed. Error estimates are obtained in Sobolev norms. Especially, the generalized Fourier transforms for some elementary polynomials and their convergence are investigated.

Normalized Solutions of Schrödinger Equations with Potentially Bounded Measures

Given a potentially bounded signed measure μ on a Brelot space (X,ℋ) with Green function G, it is well known that μ-harmonic functions (i.e., in the classical case, finely continuous versions of solutions to Δu−uμ=0) may be very discontinuous. In this paper it is shown that under very general assumptions on G (satisfied for large classes of elliptic second-order linear differential operators) normalized perturbation, however, leads to a Brelot space (X, $$\widetilde{\mathcal{H}}$$ μ) admitting a Green function T μ(G) which is locally (or even globally) comparable with G and has all properties required of G before. In particular, iterated perturbation is possible. Moreover, intrinsic Holder continuity of quotients of harmonic functions with respect to the local quasimetric ρ:=(G −1+* G −1)/2 yields ρ-Holder continuity for quotients of μ-harmonic functions as well.

Estimates of Integral Kernels for Semigroups Associated with Second-Order Elliptic Operators with Singular Coefficients

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.

Discrete Spectrum of Differential Operators in Spectral Gaps under Nonnegative Perturbations of High Order

Let A be a self-adjoint elliptic second-order differential operator, let (α, β) be an inner gap in the spectrum of A, and let B(t) = A + tW * W, where W is a differential operator of higher order. Conditions are obtained under which the spectrum of the operator B(t) in the gap (α, β) is either discrete, or does not accumulate to the right-hand boundary of the spectral gap, or is finite. The quantity N(λ, A, W, τ), λ ∈ (α, β), τ > 0 (the number of eigenvalues of the operator B(t) passing the point λ ∈ (α, β) as t increases from 0 to τ) is considered. Estimates of N(λ, A, W, τ) are obtained. For the perturbation W * W of a special form, the asymptotics of N(λ, A, W, τ) as τ → +∞ is given. Bibliography: 5 titles.

Half-eigenvalues of elliptic operators

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).

Bounded holomorphic functional calculus for non-divergence form differential operators

Let $L$ be a second-order elliptic partial differential operator of non-divergence form acting on ${\bf R^n}$ with bounded coefficients. We show that for each $1 < p_0 <2, L$ has a bounded $H_{\infty}$-functional calculus on $L^p({\bf R^n})$ for $p_0 <p <\infty$ if the $BMO$ norm of the coefficients is sufficiently small.

Heat kernel approach in quantum field theory

We give a short overview of the effective action approach in quantum field theory and quantum gravity and describe various methods for calculation of the asymptotic expansion of the heat kernel for second-order elliptic partial differential operators acting on sections of vector bundles over a compact Riemannian manifold. We consider both Laplace type operators and non-Laplace type operators on manifolds without boundary as well as Laplace type operators on manifolds with boundary with oblique and non-smooth boundary conditions.

HEAT KERNEL ASYMPTOTICS OF OPERATORS WITH NON-LAPLACE PRINCIPAL PART

We consider second-order elliptic partial differential operators acting on sections of vector bundles over a compact Riemannian manifold without boundary, working without the assumption of Laplace-like principal part -∇μ∇μ. Our objective is to obtain information on the asymptotic expansions of the corresponding resolvent and the heat kernel. The heat kernel and the Green's function are constructed explicitly in the leading order. The first two coefficients of the heat kernel asymptotic expansion are computed explicitly. A new semi-classical ansatz as well as the complete recursion system for the heat kernel of non-Laplace type operators is constructed. Some particular cases are studied in more detail.

Abel-Lidskii bases in the non-self-adjoint inverse boundary problem

Let M be a manifold with boundary δM≠Ф. Let A be a second-order elliptic partial differential operator given on M. Denote by Rλ(x, y), x, y∈M, λε\(\mathbb{C}\)\σ(A) the Schwartz kernel of (A−λI)−1. Consider the Gel'fand inverse boundary problem of the reconstruction of (M, A) via a given Rλ(x, y), x, y∈δM,λε\(\mathbb{C}\). We prove that if the principal symbol of A satisfies some geometric condition (the Bardos-Lebeau-Rauch condition), then these data determine M uniquely, and they determine A to within the group of generalized gauge transformation on M. The above-mentioned geometric condition means, roughly speaking, that any geodesic (in the metric generated by A) leaves M. Bibliography: 29 titles.

Growing-Up Positive Solutions of Semilinear Elliptic Boundary Value Problems

This paper is devoted to the study of the existence, uniqueness, and asymptotic behavior of positive solutions of a class of degenerate boundary value problems for semilinear second-order elliptic differential operators which originates from the so-called Yamabe problem in Riemannian geometry. Our approach is based on the super-sub-solution method adapted to the degenerate case.