Biharmonic Green Function Research Articles

BIHARMONIC GREEN FUNCTION AND BISUPERMEDIAN ON INFINITE NETWORKS

In this article, we have discussed Biharmonic Green function on an infinite network and bimedian functions. We have proved some standard results in terms of supermedian and bimedian. Also, we have proved the Discrete Riquier problem in the setting of bimedian functions.

An elementary proof that the triharmonic Green function of an eccentric ellipse changes sign

The conjecture named after Boggio and Hadamard that a biharmonic Green function on convex domains is of fixed sign is known to be false. One might ask what happens for the triharmonic Green function on convex domains. On disks and balls it is known to be positive. We will show that also this Green function is not positive on some eccentric ellipse.

Site 2D Velocity Structure Imaged by Microtremors Method

In this paper, a two-step space prediction approach is developed to image site 2D velocity structures. Firstly the average Vs of each layer are fixed, the interfaces between each two soils layers are predicted by the interpolation method of bi-harmonic Green function as spline in tension, to keep soil interfaces varying smoothly in space. Furthermore, a method to determine tension coefficient is proposed based on microtremor H/V method, which makes most use of H/V information and avoids choosing the tension coefficient at will. Secondly, the variance essence of the Vs structure as a regional variable is deeply excavated by Kriging method and its structural characteristic is taken into account. The spatial successive distribution of the Vs structure between soil interfaces is simulated by high precision space extrapolation and prediction, and finally the detailed 2D model is constructed. The two-step space prediction approach describes the characteristic of Vs structure as a regional variable from the sparsely and non-uniformly distributed microtremors data. Obviously, it also can easily make good use of the other exploration data.

Biharmonic Green function and biharmonic Neumann function in a sector

In this article, a biharmonic Green function and a biharmonic Neumann function in a sector with angle θ = π/n (n ∈ ℕ) are studied, and then we investigate the related Dirichlet problem and Neumann problem for the bi-Poisson equation, respectively.

Optimal estimates from below for biharmonic Green functions

Optimal pointwise estimates are derived for the biharmonic Green function under Dirichlet boundary conditions in arbitrary C 4,γ-smooth domains. Maximum principles do not exist for fourth order elliptic equations and the Green function may change sign. It prevents using a Harnack inequality as for second order problems and hence complicates the derivation of optimal estimates. The present estimate is obtained by an asymptotic analysis. The estimate shows that this Green function is positive near the singularity and that a possible negative part is small in the sense that it is bounded by the product of the squared distances to the boundary.

An Integral Formula in Weighted Bergman Spaces

In order to establish that extremal functions in the Bergman space A p act as both expansive multipliers and contractive divisors, Duren, Khavinson, Shapiro and Sundberg made use of an integral formula involving the biharmonic Green function. Using a weighted biharmonic Green function, we derive an analogous integral formula in the standard weighted Bergman space \(A_{\alpha}^{p}\ {\rm when}\ \alpha =1\), and we also discuss how the formula can be established for general α. Moreover, we show that each \(A_{\alpha}^{p}\) inner function acts as a contractive divisor on the invariant subspace which it generates.

On a biharmonic problem in a circular ring

Several problems of plane elasticity theory which can be reduced to different biharmonic boundary value problems are described, and the concept of biharmonic Green functions is discussed. The explicit formula for the particular biharmonic Green function in a circular ring R = { z ∈ C: 0< r <| z |<1} is presented and applied to the solution of the Dirichlet boundary value problem for the bi-Poisson equation (∂ z ∂ z¯ ) 2 w = f in R , w = γ 0 , ;∂ z ∂ z¯ w = γ 1 on ∂ R , for given f ∈ L p ( R ;C), p >2, γ 0 , γ 1 ∈ C (∂ R ;C).

An expansive multiplier property for operator-valued Bergman inner functions

We show that operator-valued Bergman inner functions have the so-called expansive multiplier property generalizing a well-known result of Hedenmalm in the scalar case. This analysis leads to norm bounds for input output maps for a related class of discrete time linear systems. The proof uses properties of the biharmonic Green function (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Biharmonic Green Functions on Homogeneous Trees

The study of biharmonic functions under the ordinary (Euclidean) Laplace operator on the open unit disk $${\mathbb{D}}$$ in $${\mathbb{C}}$$ arises in connection with plate theory, and in particular, with the biharmonic Green functions which measure, subject to various boundary conditions, the deflection at one point due to a load placed at another point. A homogeneous tree T is widely considered as a discrete analogue of the unit disk endowed with the Poincare metric. The usual Laplace operator on T corresponds to the hyperbolic Laplacian. In this work, we consider a bounded metric on T for which T is relatively compact and use it to define a flat Laplacian which plays the same role as the ordinary Laplace operator on $${\mathbb{D}}$$ . We then study the simply-supported and the clamped biharmonic Green functions with respect to both Laplacians.

Boundedness of the negative part of biharmonic Green's functions under Dirichlet boundary conditions in general domains

In general, higher order elliptic equations and boundary value problems like the biharmonic equation or the linear clamped plate boundary value problem do not enjoy neither a maximum principle nor a comparison principle or – equivalently – a positivity preserving property. It is shown that, on the other hand, for bounded smooth domains Ω ⊂ R n , the negative part of the corresponding Green's function is “small” when compared with its singular positive part, provided that n ⩾ 3 . To cite this article: H.-Ch. Grunau, F. Robert, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Boundedness of the gradient of a solution and Wiener test of order one for the biharmonic equation

The behavior of solutions to the biharmonic equation is well-understood in smooth domains. In the past two decades substantial progress has also been made for the polyhedral domains and domains with Lipschitz boundaries. However, very little is known about higher order elliptic equations in the general setting. In this paper we introduce new integral identities that allow to investigate the solutions to the biharmonic equation in an arbitrary domain. We establish: (1) boundedness of the gradient of a solution in any three-dimensional domain; (2) pointwise estimates on the derivatives of the biharmonic Green function; (3) Wiener-type necessary and sufficient conditions for continuity of the gradient of a solution.

On the integral representation of superbiharmonic functions

We consider a nonnegative superbiharmonic function w satisfying some growth condition near the boundary of the unit disk in the complex plane. We shall find an integral representation formula for w in terms of the biharmonic Green function and a multiple of the Poisson kernel. This generalizes a Riesz-type formula already found by the author for superbihamonic functions w satisfying the condition 0 ⩽ w(z) ⩽ C(1-|z|) in the unit disk. As an application we shall see that the polynomials are dense in weighted Bergman spaces whose weights are superbiharmonic and satisfy the stated growth condition near the boundary.

Regions of positivity for polyharmonic Green functions in arbitrary domains

The Green function for the biharmonic operator on bounded domains with zero Dirichlet boundary conditions is in general not of fixed sign. However, by extending an idea of Z. Nehari, we are able to identify regions of positivity for Green functions of polyharmonic operators. In particular, the biharmonic Green function is considered in all space dimensions. As a consequence we see that the negative part of any such Green function is somehow small compared with the singular positive part.

Numerical differentiation for the second order derivatives of functions of two variables

A regularized optimization problem for computing numerical differentiation for the second order derivatives of functions with two variables from noisy values at scattered points is discussed in this article. We prove the existence and uniqueness of the solution to this problem, provide a constructive scheme for the solution which is based on bi-harmonic Green's function and give a convergence estimate of the regularized solution to the exact solution for the problem under a simple choice of regularization parameter. The efficiency of the constructive scheme is shown by some numerical examples.

A monotonicity estimate of the biharmonic Green function

In this note we present a refined form of the monotonicity property for the biharmonic Green function and the related so called harmonic compensator. Specialized to the harmonic compensator the estimate obtained is shown to be sharp.

Construction of Green functions for weighted biharmonic operators

A constructive proof is given of the existence of the weighted biharmonic Green function Γ α for α⩾0. The method is used to derive the explicit formula for Γ 1 previously stated by Hedenmalm. In addition, a formula for Γ 2 is found, which is then shown to take both positive and negative values in the bidisk D× D .

Application of semigroups to biharmonic Green functions

Application of semigroups to biharmonic Green functions

Bergman-Type Reproducing Kernels, Contractive Divisors, and Dilations

Let Ω be a region in the complex plane. In this paper we introduce a class of sesquianalytic reproducing kernels on Ω that we call B-kernels. When Ω is the open unit disk D and certain natural additional hypotheses are added we call such kernels k Bergman-type kernels. In this case the associated reproducing kernel Hilbert space H(k) shares certain properties with the classical Bergman space L2α of the unit disk. For example, the weighted Bergman kernels kβw(z)=(1−wz)−β, 1⩽β⩽2 are Bergman-type kernels. Furthermore, for any Bergman-type kernel k one has H2⊆H(k)⊆L2a, where the inclusion maps are contractive, and Mζ, the operator of multiplication with the identity function ζ, defines a contraction operator on H(k). Our main results about Bergman-type kernels k are the following two: First, once properly normalized, the reproducing kernel for any nontrivial zero based invariant subspace M of H(k) is a Bergman-type kernel as well. For the weighted Bergman kernels kβ this result even holds for all Mζ-invariant subspace M of index 1, i.e., whenever the dimension of M/ζM is one. Second, if M is any multiplier invariant subspace of H(k), and if we set C∗=M⊖zM, then Mζ∣M is unitarily equivalent to Mζ acting on a space of C∗-valued analytic functions with an operator-valued reproducing kernel of the typekw(z)=(IC∗−zwV(z)V(w)*)kw(z), where V is a contractive analytic function V:D→L(E, C∗), for some auxiliary Hilbert space E. Parts of these theorems hold in more generality. Corollaries include contractive divisor, wandering subspace, and dilation theorems for all Bergman-type reproducing kernel Hilbert spaces. When restricted to index one invariant subspaces of H(kβ), 1⩽β⩽2, our approach yields new proofs of the contractive divisor property, the strong contractive divisor property, and the wandering subspace theorems and inner–outer factorization. Our proofs are based on the properties of reproducing kernels, and they do not involve the use of biharmonic Green functions as had some of the earlier proofs.

The harmonic Bergman kernel and the Friedrichs operator

The harmonic Bergman kernel QΩ for a simply, connected planar domain Ω can be expanded in terms of powers of the Friedrichs operator FΩ ║FΩ║<1 in operator norm. Suppose that Ω is the image of a univalent analytic function ø in the unit disk with ø' (z)=1+ψ (z) where ψ(0)=0. We show that if the function ψ belongs to a space Ds (D), s>0, of Dirichlet type, then provided that ║ψ║∞<1 the series for QΩ also converges pointwise in $\bar \Omega \times \bar \Omega \backslash \Delta (\partial \Omega )$ , and the rate of convergence can be estimated. The proof uses the eigenfunctions of the Friedrichs operator as well as a formula due to Lenard on projections in Hilbert spaces. As an application, we show that for every s>0 there exists a constant Cs>0 such that if ║ψ║Ds(D)≤Cs, then the biharmonic Green function for Ω=ø (D) is positive.

Weighted biharmonic green functions for rational weights

We present an algorithm for computing the Green function of the weighted biharmonic operator Δ|P′|−2Δ on the unit disc (with Dirichlet boundary conditions) for rational functions P. As an application, we show that if P is a Blaschke product with two zeros α1, α2 the Green function is positive if and only if |(α1−α2)/(1−{\bar α}1α2)|≤{2 \over 7}{\sqrt 10}, and also obtain an explicit formula for the Green function of the operator Δ|G|−2Δ, where G is the canonical zero-divisor of a finite zero set on the Bergman space.