Elliptic Partial Differential Operators Research Articles

Weyl Transforms, the Heat Kernel and Green Function of a Degenerate Elliptic Operator

We give a formula for the heat kernel of a degenerate elliptic partial differential operator L on ℝ2 related to the Heisenberg group. The formula is derived by means of pseudo-differential operators of the Weyl type, {i.e.}, Weyl transforms, and the Fourier–Wigner transforms of Hermite functions, which form an orthonormal basis for L2(ℝ2). Using the heat kernel, we give a formula for the Green function of L. Applications to the global hypoellipticity of L in the sense of tempered distributions, the ultracontractivity and hypercontractivity of the strongly continuous one-parameter semigroup e−tL, t > 0, are given.

A geometric theory for preconditioned inverse iteration IV: On the fastest convergence cases

In order to compute the smallest eigenvalue together with an eigenfunction of a self-adjoint elliptic partial differential operator one can use the preconditioned inverse iteration scheme, also called the preconditioned gradient iteration. For this iterative eigensolver estimates on the poorest convergence have been published by several authors. In this paper estimates on the fastest possible convergence are derived. To this end the convergence problem is reformulated as a two-level constrained optimization problem for the Rayleigh quotient. The new convergence estimates reveal a wide range between the fastest possible and the slowest convergence.

Complex symplectic spaces and boundary value problems

This paper presents a review and summary of recent research on the boundary value problems for linear ordinary and partial differential equations, with special attention to the investigations of the current authors emphasizing the applications of complex symplectic spaces. In the first part of the previous century, Stone and von Neumann formulated the theory of self-adjoint extensions of symmetric linear operators on a Hilbert space; in this connection Stone developed the properties of self-adjoint differential operators generated by boundary value problems for linear ordinary differential equations. Later, in diverse papers, Glazman, Krein and Naimark introduced certain algebraic techniques for the treatment of appropriate generalized boundary conditions. During the past dozen years, in a number of monographs and memoirs, the current authors of this expository summary have developed an extensive algebraic structure, complex symplectic spaces, with applications to both ordinary and partial linear boundary value problems. As a consequence of the use of complex symplectic spaces, the results offer new insights into the theory and use of indefinite inner product spaces, particularly Krein spaces, from an algebraic viewpoint. For instance, detailed information is obtained concerning the separation and coupling of the boundary conditions at the endpoints of the intervals for ordinary differential operators (see the Balanced Intersection Principle), and the introduction of the generalized boundary conditions over the region for some elliptic partial differential operators (see the Harmonic operator).

On a Liouville-type comparison principle for solutions of semilinear elliptic partial differential inequalities

This Note is devoted to the study of a Liouville-type comparison principle for entire weak solutions of semilinear elliptic partial differential inequalities of the form L u + | u | q − 1 u ⩽ L v + | v | q − 1 v , where q > 0 is a given number and L is a linear (possibly non-uniformly) elliptic partial differential operator of second order in divergent form given formally by the relation L = ∑ i , j = 1 n ∂ ∂ x i [ a i j ( x ) ∂ ∂ x j ] . We assume that n ⩾ 2 , that the coefficients a i j ( x ) , i , j = 1 , … , n , are measurable bounded functions on R n such that a i j ( x ) = a j i ( x ) , and that the corresponding quadratic form is non-negative. The results obtained in this work complete similar results on solutions of quasilinear elliptic partial differential inequalities announced in Kurta [C. R. Acad. Sci. Paris, Ser. I 336 (11) (2003) 897–900]. To cite this article: V.V. Kurta, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

The strengthened C.B.S. inequality constant for second order elliptic partial differential operator and for hierarchical bilinear finite element functions

We estimate the constant in the strengthened Cauchy-Bunyakowski-Schwarz inequality for hierarchical bilinear finite element spaces and elliptic partial differential equations with coefficients corresponding to anisotropy (orthotropy). It is shown that there is a nontrivial universal estimate, which does not depend on anisotropy. Moreover, this estimate is sharp and the same as for hierarchical linear finite element spaces.

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.

Non-Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators

This paper deals with the construction of a class of non-Gaussian positive-definite matrix-valued random fields whose mathematical properties allow elliptic stochastic partial differential operators to be modeled. The properties of this class is studied in details and the numerical procedure for constructing numerical realizations of the trajectories is explicitly given. Such a matrix-valued random field can directly be used for modeling random uncertainties in computational sciences with a stochastic model having a small number of parameters. The class of random fields which can be approximated is presented and their experimental identification is analyzed. An example is given in three-dimensional linear elasticity for which the fourth-order elasticity tensor-valued random field is constructed for a random non-homogeneous anisotropic elastic material.

On some stochastic parabolic differential equations in a Hilbert space

We consider some stochastic difference partial differential equations of the form du(x, t, c) = L(x, t, D)u(x, t, c)dt + M(x, t, D)u(x, t − a, c)dw(t), where L(x, t, D) is a linear uniformly elliptic partial differential operator of the second order, M(x, t, D) is a linear partial differential operator of the first order, and w(t) is a Weiner process. The existence and uniqueness of the solution of suitable mixed problems are studied for the considered equation. Some properties are also studied. A more general stochastic problem is considered in a Hilbert space and the results concerning stochastic partial differential equations are obtained as applications.

On isomorphism of ordinary differential operators

The article deals with an ordinary linear differential operator L of even order 2m with constant coefficients which defines a natural mapping of the space to itself. The operator L is considered under an extra condition that its characteristic polynomial has no real roots and exactly m roots with strictly positive imaginary part. This work prepares the treatment of properly elliptic partial differential operators in bounded domains of It turns out that there exists an extension operator to L which is an isomorphism between the standard Sobolev spaces Hm ( a , b ) and and such that the inverse operator can be represented in an extremely simple form using the Fourier transform.

On harmonic and biharmonic Bézier surfaces

We present a new method of surface generation from prescribed boundaries based on the elliptic partial differential operators. In particular, we focus on the study of the so-called harmonic and biharmonic Bézier surfaces. The main result we report here is that any biharmonic Bézier surface is fully determined by the boundary control points. We compare the new method, by way of practical examples, with some related methods such as surfaces generation using discretisation masks and functional minimisations.

On the formula of Jacques-Louis Lions for reproducing kernels of harmonic and other functions

Article On the formula of Jacques-Louis Lions for reproducing kernels of harmonic and other functions was published on June 1, 2004 in the journal Journal für die reine und angewandte Mathematik (volume 2004, issue 570).

On a Liouville-type comparison principle for solutions of quasilinear elliptic inequalities

We characterize in terms of monotonicity basic properties of quasilinear elliptic partial differential operators which make it possible to obtain a Liouville-type comparison principle for entire solutions of quasilinear elliptic partial differential inequalities of the form A( u)+| u| q−1 u⩽ A( v)+| v| q−1 v, which belong only locally to the corresponding Sobolev spaces on R n, n⩾2 . We establish that such properties are inherent for a wide class of quasilinear elliptic partial differential operators. Typical examples of such operators are the p-Laplacian and its well-known modifications for 1< p⩽2. To cite this article: V.V. Kurta, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

On the Cauchy problem for some fractional order partial differential equations

In the present paper, we study the Cauchy problem for a fractional order partial differential equation of the form D t α ∂u(x,t) ∂t −∑ |q|=2ma q(x,t)D x qu(x,t) =∑ |q|<2mb q(x,t)D x qu(x,t), where 0< α⩽1 and ∑ | q|=2 m a q ( x, t) D x q is a uniformly elliptic partial differential operator. The existence and uniqueness of the solution of the considered Cauchy problem is studied, also a continuity property is considered.

Elliptic partial differential operators and symplectic algebra

Introduction: Organization of results Review of Hilbert and symplectic space theory GKN-theory for elliptic differential operators Examples of the general theory Global boundary conditions: Modified Laplace operators Appendix A. List of symbols and notations Bibliography Index.

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).

On the fundamental solutions of a class of elliptic quartic operators in dimension 3

The (uniquely determined) even and homogeneous fundamental solutions of the linear elliptic partial differential operators with constant coefficients of the form ∑ j=1 3∑ k=1 3 c jk ∂ j 2 ∂ k 2 are represented by elliptic integrals of the first kind. Thereby we generalize Fredholm's example ∂ 1 4+ ∂ 2 4+ ∂ 3 4, which, up to now, was the only irreducible homogeneous quartic operator in 3 variables the fundamental solution of which was known to be expressible by tabulated functions.

Approximation on Closed Sets by Analytic or Meromorphic Solutions of Elliptic Equations and Applications

AbstractGiven a homogeneous elliptic partial differential operator L with constant complex coefficients and a class of functions (jet-distributions) which are defined on a (relatively) closed subset of a domain Ω in Rn and which belong locally to a Banach space V, we consider the problem of approximating in the norm of V the functions in this class by “analytic” and “meromorphic” solutions of the equation Lu = 0. We establish new Roth, Arakelyan (including tangential) and Carleman type theorems for a large class of Banach spaces V and operators L. Important applications to boundary value problems of solutions of homogeneous elliptic partial differential equations are obtained, including the solution of a generalized Dirichlet problem.

Uniform L1 error bounds for the semidiscrete solution of a Volterra equation with completely monotonic convolution kernel

We study the second-order continuous time Galerkin approximation of the solution u t + ∫ 0 t β( t — s) Au( s) ds = 0, u(0) = v, t > 0, where A is an elliptic partial-differential operator and β( t) is completely monotonic and locally integrable, but not constant. Error estimates are derived in the norm of L t 1(0, ∞; L x 2).

Extension phenomena in multidimensional complex analysis: correction of the historical record

Even though Bochner should not be credited with the proof of any version of theCR extension theorem, his 1943 paper remains a landmark in the history of the Hartogs extension phenomenon. His vision to enlarge his horizon from holomorphic functions to certain harmonic functions set the stage for further generalizations by himself (for example [Bochner 1954]) as well as for Ehrenpreis’s investigations on related problems for solutions of more general elliptic partial-differential operators.

Homotopic Classification of Euler-Lagrange Systems

In this paper we examine the linear elliptic partial differential operators that appear as Euler-Lagrange systems of certain variational integrals. We give a sufficient condition for those systems to be deformable to the Laplace system.