Elliptic Partial Differential Operators Research Articles

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.

Asymptotic analysis of a kernel estimator for parabolic SPDE's with time-dependent coefficients

In this paper we construct a kernel estimator of a time-varying coefficient of a strongly elliptic partial differential operator in a stochastic parabolic equation.The equation is assumed diagonalizable; that is, all the operators have a common system of eigenfunctions.The mean-square convergence of the estimator is established. The rate of convergence is determined both by the smoothness of the true coefficient and by the asymptotics of the eigenvalues of the operators in the equation.

Asymptotic Analysis of the Sieve Estimator for a Class of Parabolic SPDEs

In this paper we consider the problem of estimating a coefficient of a strongly elliptic partial differential operator in stochastic parabolic equations. The coefficient is a bounded function of time. We compute the maximum likelihood estimate of the function on an approximating space (sieve) using a finite number of the spatial Fourier coefficients of the solution and establish conditions that guarantee consistency and asymptotic normality of the resulting estimate as the number of the coefficients increases. The equation is assumed diagonalizable in the sense that all the operators have a common system of eigenfunctions.

Dense Subsets of [formula omitted]1-Solutions to Linear Elliptic Partial Differential Equations

Let Ω⊂RN (N⩾2) be an unbounded domain, and Lm be a homogeneous linear elliptic partial differential operator with constant coefficients. In this paper we show, among other things, that rapidly decreasing L1-solutions to Lm (in Ω) approximate all L1-solutions to Lm (in Ω), provided there exist real numbers Rj→∞, ε⩾0, and a sequence {yj} such that B(yj, ε)∩Ω=∅ and|Λ(yj, Rj, RN\\Ω)|RNj>ε ∀ j,where |·| means the volume andΛ(z, R, D)≔∪x∈B(z, R)∩Dz+t(x−z)|x−z|;t⩽1,for z∈RN, R>0 and D⊂RN. For m=2, we can replace the volume density by the capacity-density. It appears that the problem is related to the characterization of largest sets on which a nonzero polynomial solution to Lm may vanish, along with its (m−1)-derivatives. We also study a similar approximation problem for polyanalytic functions in C.

COVARIANT TECHNIQUES FOR COMPUTATION OF THE HEAT KERNEL

The heat kernel associated with an elliptic second-order partial differential operator of Laplace type acting on smooth sections of a vector bundle over a Riemannian manifold, is studied. A general manifestly covariant method for computation of the coefficients of the heat kernel asymptotic expansion is developed. The technique enables one to compute explicitly the diagonal values of the heat kernel coefficients, so called Hadamard–Minakshisundaram–De Witt–Seeley coefficients, as well as their derivatives. The elaborated technique is applicable for a manifold of arbitrary dimension and for a generic Riemannian metric of arbitrary signature. It is very algorithmic, and well suited to automated computation. The fourth heat kernel coefficient is computed explicitly for the first time. The general structure of the heat kernel coefficients is investigated in detail. On the one hand, the leading derivative terms in all heat kernel coefficients are computed. On the other hand, the generating functions in closed covariant form for the covariantly constant terms and some low-derivative terms in the heat kernel coefficients are constructed by means of purely algebraic methods. This gives, in particular, the whole sequence of heat kernel coefficients for an arbitrary locally symmetric space.

Continuous variation of eigenvalues and Gärding's inequality

Let D t be a smooth 1-parameter family of elliptic self-adjoint partial differential operators on a compact manifold without boundary. We use Gärding's inequality to give a simple proof that the kth eigenvalue of D t is continuous in t.

On radial limit functions for entire solutions of second order elliptic equations in $\mathbf{R}^2$

Given a homogeneous elliptic partial differential operator $L$ of order two with constant complex coefficients in $\bold R^2$, we consider entire solutions of the equation $Lu=0$ for which $$ \lim_{r \rightarrow \infty} u (re^{i\varphi}) =: U (e^{i\varphi}) $$ exists for all $\varphi \in [0, 2\pi)$ as a finite limit in $\bold C$. We characterize the possible radial limit functions $U$. This is an analog of the work of A. Roth for entire holomorphic functions. The results seem new even for harmonic functions.

Multigrid smoothing for symmetric nine-point stencils

Analytical formulae are obtained for the smoothing factors yielded by damped Jacobi relaxation and by red-black relaxation applied to symmetric nine-point stencil discretizations of elliptic partial differential operators in 2D. The results include point and line relaxation and full and partial coarsening. Several unusual results are implied by the formulae. Numerical results of multigrid cycles match the analytical predictions well.

Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions

For a homogeneous elliptic partial differential operator with constant coefficients and a class of functions (jet-distributions) defined on a closed, not necessarily compact, subset of and belonging locally to a Banach space , the approximation in the norm of of functions in this class by entire and meromorphic solutions of the equation is considered. Theorems of Runge, Mergelyan, Roth, and Arakelyan type are established for a wide class of Banach spaces and operators they encompass most of the previously considered generalizations of these theorems but also include new results.

A Bott Periodicity Theorem for Infinite Dimensional Euclidean Space

We formulate and prove an equivariant Bott periodicity theorem for infinite dimensional Euclidean vector spaces. The main features of our argument are (i) the construction of a non-commutativeC*-algebra to play the role of the algebra of functions on infinite dimensional Euclidean space; and (ii) the construction of a certain index one elliptic partial differential operator which provides the basis for an inverse to the Bott periodicity map. These tools have applications to index theory and the Novikov conjecture, notably a proof of the Novikov conjecture for amenable groups (the applications will be considered elsewhere).

Functional calculi of second-order elliptic partial differential operators with bounded measurable coefficients

Consider a second-order elliptic partial differential operatorL in divergence form with real, symmetric, bounded measurable coefficients, under Dirichlet or Neumann conditions on the boundary of a strongly Lipschitz domain Ω. Suppose that 1 0. ThenL has a bounded H∞ functional calculus in Lp(Ω), in the sense that ¦¦f (L +cI)u¦¦p ≤C sup¦arλ¦<μ¦f¦ ¦‖u¦‖p for some constantsc andC, and all bounded holomorphic functionsf on the sector ¦ argλ¦ < μ that contains the spectrum ofL +cI. We prove this by showing that the operatorsf(L + cI) are Calderon-Zygmund singular integral operators.

Quantitative Approximation Theorems for Elliptic Operators

LetL(D) be an elliptic linear partial differential operator with constant coefficients and only highest order terms. For compact setsK⊂RNwhose complements are John domains we prove a quantitative Runge theorem: if a functionfsatisfiesL(D)f=0 on a fixed neighborhood ofK, we estimate the sup-norm distance fromfto the polynomial solutions of degree at mostn. The proof utilizes a two-constants theorem for solutions to elliptic equations. We then deduce versions of Jackson and Bernstein theorems for elliptic operators.

On Iterations of the Green Integrals and their Applications to Elliptic Differential Complexes

AbstractConvergence of iterations of special Green integrals for overdetermined elliptic linear partial differential operators P of order p ≥ 1 is proved. Using this result we obtain necessary and sufficient conditions for the solvability of the equation Pu = f in 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 a P‐Neumann problem for elliptic differential operators is proved.

Intermittency-Type Estimates for Some Nondegenerate SPDE'S

In this paper we prove some intermittency-type estimates for the stochastic partial differential equation $du = \mathscr{L}u dt + \mathscr{M}_lu\circ dW^l_t$, where $\mathscr{L}$ is a strongly elliptic second-order partial differential operator and the $\mathscr{M}_l$'s are first-order partial differential operators. Here the $W^l$'s are standard Wiener processes and $\circ$ denotes Stratonovich integration. We assume for simplicity that $u(0,\cdot) \equiv 1$. Our interest here is the behavior of $\mathbb{E}\lbrack|u(t,x)|^p\rbrack$ for large time and large $p$; more specifically, our interest is the growth of $(p^2t)^{-1}\log\mathbb{E}\lbrack|u(t,x)|^p\rbrack$ as $t$, then $p$, become large.

Uniformly Elliptic PDEs with Bounded, Measurable Coefficients

Let \(L[\,\cdot\,]\) be a nondivergent linear second-order uniformly elliptic partial differential operator defined on functions with domain \(\Omega.\) Consider the question, "When is a function u a solution of \(L[u] = 0\) on \(\Omega\)?" The naive answer, "u is a solution of \(L[u] = 0\) on \(\Omega\) if \(u\in C^2(\Omega)\) and \(L[u](x) = 0\) for all \(x\in\Omega,\)" is clearly too limited. Indeed, if the coefficients of L are in \(W^{1,2}\cap L^{\infty},\) then L can be rewritten in divergence form for which the notion of a "weak" solution can be applied. In this case there could be infinitely many functions that are "weak" but not classical solutions. More importantly, even if the coefficients of L are just bounded and measurable, the recent results of Krylov permit us to construct "solutions" of \(L[u] = 0\) on \(\Omega,\) and these "solutions" are generally no better than continuous; the "weak" solutions previously mentioned can be obtained by this construction, too. The preceding discussion provides us with an adequate extrinsic definition of solution (i.e., given a function u we either prove that it is or is not the result of such a construction) that has been used by several authors, but one that is not particularly satisfying or illuminating. Our major contribution in this paper is to show the following. I. There is an intrinsic definition of solution that is equivalent to the extrinsic one. II. Furthermore, the intrinsic definition is just the (now) well-known Crandall-Lions viscosity solution, modified in a natural way to accommodate measurable coefficients.

Bernstein Theorems for Elliptic Equations

We study solutions of the equation L(D) u = 0, where L(D) is an elliptic linear partial differential operator with constant coefficients and only highest order terms. For compact sets K ⊂ RN with connected complement we prove a Bernstein theorem: if a function ƒ on K can be extended to a solution of the equation on an open neighborhood of K, then the supnorm distance from ƒ to the polynomial solutions of degree ≤ n decays exponentially with n. We give two proofs: a proof by duality which makes use of the theory of functions of several complex variables, and an elementary constructive proof using generalized Laurent expansions for solutions of elliptic equations. Finally, we discuss the use of orthogonal polynomial expansions, and the use of interpolation schemes, for the construction of polynomial approximations with asymptotically optimal behavior.

Numerical solution of an evolution equation with a positive-type memory term

AbstractWe study the numerical solution of an initial-boundary value problem for a Volterra type integro-differential equation, in which the integral operator is a convolution product of a positive-definite kernel and an elliptic partial-differential operator. The equation is discretised in space by the Galerkin finite-element method and in time by finite differences in combination with various quadrature rules which preserve the positive character of the memory term. Special attention is paid to the case of a weakly singular kernel. Error estimates are derived and numerical experiments reported.