Biharmonic Operator Research Articles

Universal Bounds for Eigenvalues of a Buckling Problem

In this paper, we investigate an eigenvalue problem for a biharmonic operator on a bounded domain in an n-dimensional Euclidean space, which is also called a buckling problem. We introduce a new method to construct ``nice'' trial functions and we derive a universal inequality for higher eigenvalues of the buckling problem by making use of the trial functions. Thus, we give an affirmative answer for the problem on universal bounds for eigenvalues of the buckling problem, which was proposed by Payne, Polya and Weinberger in [14] and this problem has been mentioned again by Ashbaugh in [1].

Inequalities for eigenvalues of a clamped plate problem

Let D be a connected bounded domain in an n-dimensional Euclidean space R n . Assume that 0 < λ 1 < λ 2 ≤ ··· ≤λ k ≤··· are eigenvalues of a clamped plate problem or an eigenvalue problem for the Dirichlet biharmonic operator: Then, we give an upper bound of the (k+1)-th eigenvalue λ k+1 in terms of the first k eigenvalues, which is independent of the domain D, that is, we prove the following: Further, a more explicit inequality of eigenvalues is also obtained.

Counting eigenvalues of biharmonic operators with magnetic fields

AbstractAn analysis is given of the spectral properties of perturbations of the magnetic bi‐harmonic operator Δ2A in L 2(Rn ), n = 2, 3, 4, where A is a magnetic vector potential of Aharonov–Bohm type, and bounds for the number of negative eigenvalues are established. Key elements of the proofs are newly derived Rellich inequalities for Δ2A which are shown to have a bearing on the limiting cases of embedding theorems for Sobolev spaces H 2(Rn ). (© 2005 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Asymptotic behavior of least energy solutions to a four-dimensional biharmonic semilinear problem

In this paper, we study the following fourth order elliptic problem $(E_p)$: \begin{eqnarray*} (E_p) \left \{ \begin{array}{l} \Delta^2 u = u^p \quad \mbox{in} \Omega, u > 0 \quad \mbox{in} \Omega, u |_{\partial\Omega} = \Delta u |_{\partial\Omega} = 0 \end{array} \right. \end{eqnarray*} where $\Omega$ is a smooth bounded domain in $\mathbf{R}^4$, $\Delta^2 = \Delta\Delta$ is a biharmonic operator and $p >1$ is any positive number. We investigate the asymptotic behavior as $p \to \infty$ of the least energy solutions to $(E_p)$. Combining the arguments of Ren-Wei [8] and Wei [10], we show that the least energy solutions remain bounded uniformly in $p$, and on convex bounded domains, they have one or two ``peaks'' away form the boundary. If it happens that the only one peak point appears, we further prove that the peak point must be a critical point of the Robin function of $\Delta^2$ under the Navier boundary condition.

On Homogenization of Some Boundary-Value Problems in Domains with Thin Channels

Homogenization theorems are proved for some boundary-value problems for the Laplace operator and the biharmonic operator in plane domains consisting of two or more parts joined by thin strips of small length.

Secondary flow in non-isothermal viscoelastic parallel-plate flow

The non-isothermal, parallel-plate flow of a viscoelastic fluid is investigated. Using the singular perturbation method the problem for the stream function χ is reduced to a non-homogeneous boundary-value problem for the biharmonic operator in an infinite strip. An exact solution is then obtained using Fourier integral transforms. Our analysis shows the existence of thermally induced recirculating eddies, even when inertia is neglected.

LOCATION OF THE BLOW UP POINT FOR POSITIVE SOLUTIONS OF A BIHARMONIC EQUATION INVOLVING NEARLY CRITICAL EXPONENT

In this paper a semilinear biharmonic problem involving nearly critical growth with Navier boundary condition is considered on an any bounded smooth domain. It is proved that positive solutions concentrate on a point in the domain, which is also a critical point of the Robin's function corresponding to the Green's function of biharmonic operator with the same boundary condition. Similar conclusion has been obtained in [6] under the condition that the domain is strictly convex.

Postprocessing and higher order convergence of the mixed finite element approximations of biharmonic eigenvalue problems

A new procedure for accelerating the convergence of mixed finite element approximations of the eigenpairs and of the biharmonic operator is proposed. It is based on a postprocessing technique that involves an additional solution of a source problem on an augmented finite element space. This space could be obtained either by substantially refining the grid, the two-grid method, or by using the same grid but increasing the order of polynomials by one, the two-space method. The numerical results presented and discussed in the paper illustrate the efficiency of the postprocessing method.

A Multigrid Method for a Fourth-Order Diffusion Equation with Application to Image Processing

In this paper we are concerned with the efficient solution of the elliptic system $(\alpha L + I)u = f$, where L is the biharmonic operator supplemented by higher-order boundary conditions. This is a core task in image processing arising from the minimization of rotation invariant energy functionals. For the purpose of numerical solution we approximate the partial differential equation by finite differences. In order to obtain efficient and robust solution methods we discuss how the system can be treated efficiently by multigrid methods. Finally, experimental results are reported that demonstrate the effectiveness of the multigrid approach.

Anisotropic Gent–McWilliams Parameterization for Ocean Models

Abstract An anisotropic generalization of the Gent–McWilliams (GM) parameterization is presented for eddy-induced tracer transport and diffusion in ocean models, and it is implemented in an ocean general circulation model using a functional formalism to derive the spatial discretization. This complements the anisotropic viscosity parameterization recently developed by Smith and McWilliams. The anisotropic GM operator is potentially useful in both coarse- and high-resolution ocean models, and in this study the focus is on its application in high-resolution eddying solutions, for which it provides an adiabatic alternative to the more commonly used biharmonic horizontal diffusion operators. It is shown that realistically high levels of eddy energy can be simulated using harmonic anisotropic diffusion and friction operators. Isotropic forms can also be used, but these tend either to overly damp the solution when a large diffusion coefficient is used or to introduce unacceptable levels of numerical noise when a small coefficient is used. A series of numerical simulations of the North Atlantic Ocean are conducted at 0.2° resolution using anisotropic viscosity, anisotropic GM, and biharmonic mixing operators to investigate the effects of the anisotropic forms and to isolate changes in the solutions specifically associated with anisotropic GM. A high-resolution 0.1° simulation is then conducted using both anisotropic forms, and the results are compared with a similar run using biharmonic mixing. Modest improvements are seen in the mean wind-driven circulation with the anisotropic forms, but the largest effects are due to the anisotropic GM parameterization, which eliminates the spurious diapycnal diffusion inherent in horizontal tracer diffusion. This leads to significant improvements in the model thermohaline circulation, including the meridional heat transport, meridional overturning circulation, and deep-water formation and convection in the Labrador Sea.

Advection diffusion in nonchaotic closed flows: Non-Hermitian operators, universality, and localization

The qualitative spectral properties characterizing the advection-diffusion operator in two-dimensional steady incompressible flows can be obtained from the analysis of simple model flows on the torus, the velocity field of which attains the simple expression v (x) = (0, v(y) (x) ) . For this class of simple flows, the advection-diffusion operator reduces to a one-dimensional Schrödinger operator in the presence of an imaginary potential, which shares some spectral analogies with non-Hermitian quantum operators (e.g., spectral invariance), and is characterized by eigenfunction localization. The latter property (i.e., eigenfunction localization) is strictly related to the occurrence of a universal scaling of the eigenvalue spectrum with the Peclet number, the scaling exponent of which depends exclusively on the local behavior of the potential close to its critical points. The analysis is extended to a class of unbounded non-Hermitian operators, which include the Laplacian and the biharmonic operators coupled to an imaginary potential as special cases.

Linear stability of viscoelastic cone–plate flow in a bounded domain

We analyze the elastic instability of torsional flow of a viscoelastic fluid in a cone-and-plate device with gap angle α. Assuming that the flow is axisymmetric and inertialess, we derive a model eigenvalue problem for the perturbation stream function valid for α≪1. The problem reduces to a singularly perturbed fourth-order boundary value problem related to the biharmonic operator. This problem is solved numerically and the results are compared with those obtained for radially unbounded domains. Assuming that the meniscus perturbation is small enough to preclude capillary instabilities, the flow is found to be slightly more stable than is predicted by the results for radially unbounded regions.

Biharmonic capacity and the stability of minimal Lagrangian submanifolds

We study the eigenvalues of the biharmonic operators and the buckling eigenvalue on complete, open Riemannian manifolds. We show that the first eigenvalue of the biharmonic operator on a complete, parabolic Riemannian manifold is zero. We give a generalization of the buckling eigenvalue and give applications to studying the stability of minimal Lagrangian submanifolds in Kahler manifolds. MSC 1991: 58G25, 53A10, 35P15 key words biharmonic, buckling eigenvalue, minimal Lagrangian submanifold

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 .

A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator

We prove a Sobolev inequality with remainder term for the imbedding D-m,D-2(R-N) hooked right arrowL(2N/(N-2m))(R-N), m is an element of N arbitrary, generalizing a corresponding result of Bianchi and Egnell for the case m=1. We also show that the manifold of least energy solutions u is an element of D-m,D-2 (R-N) of the equation (-Delta)(m) u=u(4m/(N-2m)) u is a nondegenerate critical manifold for the corresponding variational integral. Finally we generalize the results of J.M. Coron on the existence of solutions of equations with critical exponent on domains with nontrivial topology to the biharmonic operator.

Uniform stabilization of a one‐dimensional hybrid thermo‐elastic structure

AbstractThis paper is concerned with the stabilization of a one‐dimensional hybrid thermo‐elastic structure consisting of an extensible thermo‐elastic beam which is hinged at one end with a rigid body attached to its free end. The model takes account of the effect of stretching on bending and rotational inertia. The property of uniform stability of the energy associated with the model is asserted by constructing an appropriate Lyapunov functional for an abstract second order evolution problem. Critical use is made of a multiplier of an operator theoretic nature, which involves the fractional power A−1/2 of the bi‐harmonic operator pair A acting in the abstract evolution problem. An explicit decay rate of the energy is obtained. Copyright © 2003 John Wiley &amp; Sons, Ltd.

Numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions: an augmented Lagrangian approach

The main goal of this Note is to discuss a method for the numerical solution of the two-dimensional elliptic Monge–Ampère equation with Dirichlet boundary conditions (the E-MAD problem). This method relies on the reformulation of E-MAD as a problem of Calculus of Variation involving the biharmonic operator (or closely related operators), and then to a saddle-point formulation for a well-chosen augmented Lagrangian functional, leading to iterative methods such as Uzawa–Douglas–Rachford. The above methodology applies to problems other than E-MAD (such as the Pucci equation). The results of numerical experiments are presented. They concern the solution of E-MAD on the unit square (0,1)×(0,1); the first test problem has a known smooth closed form solution which is easily computed with optimal order of convergence. The second test problem has also a known closed form solution; the fact that this solution has the H 2(Ω) -regularity, but not the C 2( Ω) one, does not prevent optimal order of convergence. Finally, the third test problem having no smooth solution is more costly to solve and leads to discrete solutions showing negative curvature near the corners. To cite this article: E.J. Dean, R. Glowinski, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Mesh deformation using the biharmonic operator

AbstractThe use of the biharmonic operator for deforming a mesh in an arbitrary–Lagrangian–Eulerian simulation is investigated. The biharmonic operator has the advantage that two conditions can be specified on each boundary of the mesh. This allows both the position and the normal mesh spacing along a boundary to be controlled, which is important for two‐fluid interfaces and periodic boundaries. At these boundaries, we can simultaneously fix the position of the boundary and ensure that the normal mesh spacing is continuous across the boundary. In addition, results for deforming surfaces show that greater surface deformation can be tolerated when using biharmonic equations compared to approaches using second‐order partial differential equations. A final advantage is that with the biharmonic operator, the integrity of a grid in a moving boundary layer can be preserved as the boundary moves. The main disadvantage of the approach is its increased computational expense. Copyright © 2003 John Wiley &amp; Sons, Ltd.

Remarks on nonlinear biharmonic evolution equation of Kirchhofftype in noncylindrical domain

We investigate a boundary value problem for a nonlinear evolution biharmonic operator motivated by flexion of fully clamped beam in two different physical situations. In the first, the supports of the ends of the beam are fixed and in the second one, the supports of the ends of the beam have small displacements.

On a class of singular biharmonic problems involving critical exponents

This paper deals with the following class of singular biharmonic problems (P)Δ2u+V(x)|u|q−1u=|u|2∗−2u,inΩ⊂RN,u∈D2,2o(Ω),N⩾5, where 1⩽q<2∗−1, 2∗=2N/(N−4) is the critical Sobolev exponent, Δ2 denotes the biharmonic operator, Ω is open domain (not necessarily bounded, it may be equal to RN) and V is a potential that changes sign in Ω with some points of singularities in Ω. Some results on the existence of solutions are obtained by combining the Mountain Pass Theorem and Hardy inequality with some arguments used by Brézis and Nirenberg.