Biharmonic Green 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.

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.

Positivity and Almost Positivity of Biharmonic Green’s Functions under Dirichlet Boundary Conditions

In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem, neither a maximum principle nor a comparison principle or—equivalently—a positivity preserving property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem from being reasonably written as a system of second order boundary value problems. It is shown that, on the other hand, for bounded smooth domains \({\Omega \subset\mathbb{R}^n}\) , the negative part of the corresponding Green’s function is “small” when compared with its singular positive part, provided \({n\geqq 3}\) . Moreover, the biharmonic Green’s function in balls \({B\subset\mathbb{R}^n}\) under Dirichlet (that is, clamped) boundary conditions is known explicitly and is positive. It has been known for some time that positivity is preserved under small regular perturbations of the domain, if n = 2. In the present paper, such a stability result is proved for \({n\geqq 3}\) .

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).

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.

An Elementary Proof That the Biharmonic Green Function of an Eccentric Ellipse Changes Sign

P R. Garabedian showed in 1951 that the Green function for the biharmonic boundary value problem with vanishing Dirichlet data changes sign in case the domain is a sufficiently eccentric ellipse. This refuted a conjecture made by J. Hadamard in 1908. The proof of Garabedian was based on kernel functions; the present note gives an elementary proof.

Analytic continuation of certain domain functions in quadrature domains and domains bounded by ellipses

We use ideas from a previous paper by the author to study the analytic continuation of certain domain functions. We present new proofs of some results, which are more or less known, about the Green's function and certain conformal maps of quadrature domains and domains bounded by ellipses. We also obtain some new results about the biharmonic Green's function in quadrature domains.

Green's functions, bi-linear forms, and completeness of the eigenfunctions for the elastostatic strip and wedge

The bi-harmonic Green's functionG(r′,r) for the infinite strip region -1≤y≤1, -∞<x<∞, with the boundary conditionsG=∂G/∂y ony=±1, is obtained in integral form. It is shown thatG has an elegant bi-linear series representation in terms of the (Papkovich-Fadle) eigenfunctions for the strip. This representation is then used to show that any function ϕ bi-harmonic in arectangle, and satisfying the same boundary conditions asG, has a unique representation in the rectangle as an infinite sum of these eigenfunctions. For the case of the semi-infinite strip, we investigate conditions on ϕ sufficient to ensure that ϕ is exponentially small asx→∞. In particular it is proved that this is so, solely under the condition that ϕ be bounded asx→∞.

Existence of Biharmonic Green's Functions

Existence of Biharmonic Green's Functions

Green's function of the clamped punctured disk

If a thin elastic circular plate B: ∣z∣ &lt; 1 is clamped (simply supported, respectively) along its edge ∣z∣ = 1, its deflection at z ∈ B under a point load at ζ ∈ B, measured positively in the direction of the gravitational pull, is the biharmonic Green's function β(z, ζ) of the clamped plate (γ(z, ζ) of the simply supported plate, respectively). We ask: how do β(z, ζ) and γ(z, ζ) compare with the corresponding deflections β0(z, ζ) and γ0(z, ζ) of the punctured circular plate B0: 0 &lt; ∣ z ∣ &lt; 1 that is “clamped” or “simply supported”, respectively, also at the origin? We shall show that γ(z, ζ) is not affected by the puncturing, that is, γ(·, ζ) = γ0(·, ζ), whereas β(·, ζ) is:on B0 × B0. Moreover, while β((·, ζ) is of constant sign, β0(·, ζ) is not. This gives a simple counterexmple to the conjecture of Hadamard [6] that the deflection of a clampled thin elastic plate be always of constant sign:The biharmonic Gree's function of a clampled concentric circular annulus is not of constant sign if the radius of the inner boundary circle is sufficiently small.Earlier counterexamples to Hadamard's conjecture were given by Duffin [2], Garabedian [4], Loewner [7], and Szegö [9]. Interest in the problem was recently revived by the invited address of Duffin [3] before the Annual Meeting of the American Mathematical Society in 1974. The drawback of the counterexample we will present is that, whereas the classical examples are all simply connected, ours is not. In the simplicity of the proof, however, there is no comparison.

A nonexistence test for biharmonic Green's functions of clamped bodies.

A nonexistence test for biharmonic Green's functions of clamped bodies.

Computing internal viscous flow problems for the circle by integral methods

Steady two-dimensional viscous motion within a circular cylinder generated by (a) the rotation of part of the cylinder wall and (b) fluid entering and leaving through slots in the wall is considered. Studied in particular are moving-surface problems with continuous and discontinuous surface speeds, simple inflow–outflow problems and the impinging-jet problem within a circle. The analytical solutions at zero Reynolds number are given for the last two types of problem. Numerical results are obtained at various Reynolds numbers from the integral representation of the solution. At zero Reynolds number this approach involves a quadrature around the circumference of the cylinder. At other Reynolds numbers it involves an iterative–integral technique based on the use of the ‘clamped-plate’ biharmonic Green's function.

A Relation Between Two Biharmonic Green's Functions on Riemannian Manifolds

A Relation Between Two Biharmonic Green's Functions on Riemannian Manifolds

A strict inclusion related to biharmonic Green's functions of clamped and simply supported bodies

A strict inclusion related to biharmonic Green's functions of clamped and simply supported bodies

A criterion for the existence of biharmonic Green's functions

AbstractThe biharmonic Green's function of a simply supported plate is generalized to Riemannian manifolds and shown to exist if and only if the harmonic measure of the ideal boundary is square integrable.

Biharmonic Green's functions and biharmonic degeneracy.

Biharmonic Green's functions and biharmonic degeneracy.

A criterion for the existence of biharmonic Green's functions

The biharmonic Green's function of a simply supported plate is generalized to Riemannian manifolds and shown to exist if and only if the harmonic measure of the ideal boundary is square integrable.

Biharmonic Green's Functions and Quasiharmonic Degeneracy.

Biharmonic Green's Functions and Quasiharmonic Degeneracy.

Note on a biharmonic Green's function

An expression is found for the biharmonic Green's function, G(x,y;x/sub 1/,y/sub 1/), for an infinite area in the x-y plane bounded internally by a single curve. At all points of the boundary, G is found to have a zero of the second onder. (T.F.H.)