Biharmonic Operator Research Articles

Bifurcation and multiplicity results for fourth-order equations

ABSTRACTIn this paper, we prove that fourth-order equations involving biharmonic operator with superlinear growth at infinity and saddle structure near zero have three nontrivial solutions. The approach is based on a combination of bifurcation theory and minimax methods.

The integration of a biharmonic equation by an implicit scheme

The paper presents a step-by-step construction of a finite-difference scheme for a heterogeneous biharmonic equation under zero boundary conditions superimposed on the desired function and its first-order partial derivatives. The finite-difference scheme is based on a square twenty-five-point pattern and has an implicit character. On analytical grid, the error of approximation of the biharmonic operator by the difference analog and the error of approximation of boundary conditions imposed on the first order partial derivatives are calculated by the expansion of the function in the Taylor series with the remainder term in the form of a Lagrange. The boundary conditions imposed on the sought function are satisfied precisely. A finite-difference scheme approximates a boundary value problem with a second order of accuracy over the mesh step. With the help of the Maple computer algebra system the solutions of the problem for different grid steps are obtained. The dependence of the minimum function and calculation time on the number of significant digits is revealed. The optimal number of significant digits is found. The convergence rate of the numerical scheme is analyzed. The dependence of the minimum value of the function and the calculation time on the value of the grid step is established. The optimal step value is found. A three-dimensional graph of the solution and its profiles in the middle sections are constructed. The advantages of the developed finite-difference scheme are indicated. Obtained results correspond to the physical meaning of the problem and are consistent with similar numerical and approximate analytical solutions.

On a p(x)-biharmonic problem with Navier boundary condition

In this paper, we study a p(x)-biharmonic equation with Navier boundary condition \t\t\t{Δp(x)2u+a(x)|u|p(x)−2u=λf(x,u)+μg(x,u)in Ω,u=Δu=0on ∂Ω.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} \\Delta^{2}_{p(x)}u+a(x)|u|^{p(x)-2}u= \\lambda f(x,u)+\\mu g(x,u)\\quad \\text{in } \\Omega, \\\\ u=\\Delta u=0 \\quad \\text{on } \\partial\\Omega. \\end{cases} $$\\end{document} Here Omegasubsetmathbb{R}^{N} (Ngeq1) is a bounded domain with smooth boundary ∂Ω, Delta^{2}_{p(x)}u is a p(x)-biharmonic operator with p(x) in C(overline{Omega}), p(x)>1. lambda,muinmathbb{R}, ain L^{infty}(Omega) such that inf_{xinOmega}a(x)=a^{-}>0. By variational methods, we establish the results of existence and non-existence of solutions.

The asymptotics of natural oscillations of a long two-dimensional Kirchhoff plate with variable cross-section

After scaling, a long Kirchhoff plate with rigidly clamped ends and free lateral sides is described by a mixed boundary-value problem for the biharmonic operator in a thin domain with weakly curved boundary. Based on a general procedure for constructing asymptotic formulae for solutions of elliptic problems in thin domains, asymptotic expansions are derived for the eigenvalues and eigenfunctions of this problem with respect to a small parameter equal to the relative width of the plate and are also justified. In the low-frequency range of the spectrum the limiting problem is a Dirichlet problem for a fourth-order ordinary differential equation with variable coefficients, and in the mid-frequency range it is (quite unexpectedly) a Dirichlet problem for a second-order equation. The phenomenon of a boundary layer in a neighbourhood of the ends of the plate is investigated. This makes it possible to construct infinite formal asymptotic series for simple eigenvalues and the corresponding eigenfunctions, and to develop a model with enhanced precision. Asymptotic constructions for plates with periodic rapidly oscillating boundaries or for other sets of boundary conditions corresponding to mechanically reasonable ways to fix the ends of the plate are discussed. Bibliography: 46 titles.

For a class of p(x)-biharmonic operators with weights

We study the fourth order nonlinear problem with a p(x)-biharmonic operators $$\begin{aligned} \left\{ \begin{array}{lll} \Delta (|\Delta u|^{p_1(x)-2} \Delta u)+\Delta (|\Delta u|^{p_2(x)-2} \Delta u)&{}=\lambda V_1(x)|u|^{q(x)-2}u-\mu V_2(x)|u|^{\alpha (x)-2}u, \quad &{}x\in \Omega \\ u=\Delta u=0,&{}\quad &{} x\in \partial \Omega \\ \end{array}\right. \end{aligned}$$ where \(\Omega \in \mathbb {R}^{N}\) with \(N\ge 2\) is a bounded domain with smooth boundary, \(\lambda \), \(\mu \) are positive real numbers, \(p_{1}\), \(p_{2}\), q and \(\alpha \) are continuous functions on \(\overline{\Omega }\), \(V_1\) and \(V_2\) are weight functions in a generalized Lebesgue spaces \(L^{s_1(x)}(\Omega )\) and \(L^{s_2(x)}(\Omega )\) respectively such that \(V_1\) may change sign in \(\Omega \) and \(V_2\ge 0\) on \(\Omega \). We established an existence results using variational approaches and Ekeland’s variational principle.

On the stability of some isoperimetric inequalities for the fundamental tones of free plates

We provide a quantitative version of the isoperimetric inequality for the fundamental tone of a biharmonic Neumann problem. Such an inequality has been recently established by Chasman adaptingWeinberger’s argument for the corresponding second order problem. Following a scheme introduced by Brasco and Pratelli for the second order case, we prove that a similar quantitative inequality holds also for the biharmonic operator. We also prove the sharpness of both such an inequality and the corresponding one for the biharmonic Steklov problem.

Boundary Value Problem of the Operator ⊕k Related to the Biharmonic Operator and the Diamond Operator

This paper presents an alternative methodology for finding the solution of the boundary value problem (BVP) for the linear partial differential operator. We are particularly interested in the linear operator ⊕k, where ⊕k=♡k♢k, ♡k is the biharmonic operator iterated k-times and ♢k is the diamond operator iterated k-times. The solution is built on the Green’s identity of the operators ♡k and ⊕k, in which their derivations are also provided. To illustrate our findings, the example with prescribed boundary conditions is exhibited.

Asymptotics of the Deflection of a Cruciform Junction of Two Narrow Kirchhoff Plates

Two two-dimensional plates with bending described by Sophie Germain’s equation with the biharmonic operator are joined in the form of a cross with clamped ends, but with free lateral sides outside the cross core. Asymptotics of the deflection of the junction with respect to the relative width of the plates regarded as a small parameter is constructed and justified. The variational-asymptotic model includes a system of two ordinary differential equations of the fourth and second orders with Dirichlet conditions at the endpoints of the one-dimensional cross and the Kirchhoff transmission conditions at its center. They are derived by analyzing the boundary layer near the crossing of the plates and mean that the deflection and the angles of rotation at the central point are continuous and that the total bending force and the principal torques vanish. Possible generalizations of the asymptotic analysis are discussed.

A functional analytic perspective to the div-curl lemma

We present an abstract functional analytic formulation of the celebrated div-curl lemma found by F. Murat and L. Tartar. The viewpoint in this note relies on sequences for operators in Hilbert spaces. Hence, we draw the functional analytic relation of the div-curl lemma to differential forms and other sequences such as the Grad grad-sequence discovered recently by D. Pauly and W. Zulehner in connection with the biharmonic operator.

L2-norm blow-up of solutions to a fourth order parabolic PDE involving the Hessian

This paper deals with a fourth order parabolic PDE arising in the theory of epitaxial growth of crystal. We focalize the study on one open question proposed by Escudero et al. (2015) [4], that is, Lp-norm blow-up. For the initial energy J(u0)<λ16‖u0‖22, we prove the solution blows up in finite time with L2-norm, where λ1 is the least Dirichlet eigenvalue of the biharmonic operator. Moreover, the lifespan of the solution is got, and the results of this paper also generalize the results got by Xu and Zhou (2017) [12].

On the Number of Eigenvalues of the Biharmonic Operator on ℝ3 Perturbed by a Complex Potential

In this paper we study the biharmonic operator on ℝ3 perturbed by a complex potential, which decays exponentially at infinity. We obtain bounds on the total number of eigenvalues of the said operator.

P⁢(x)p(x)-biharmonic operator involving the p⁢(x)p(x)-Hardy inequality

Abstract In this work, we investigate the spectrum denoted by Λ for the p ⁢ ( x ) {p(x)} -biharmonic operator involving the Hardy term. We prove the existence of at least one non-decreasing sequence of positive eigenvalues of this problem such that sup ⁡ Λ = + ∞ {\sup\Lambda=+\infty} . Moreover, we prove that inf ⁡ Λ &gt; 0 {\inf\Lambda&gt;0} if and only if the domain Ω satisfies the p ⁢ ( x ) {p(x)} -Hardy inequality.

Recovery of singularities from a backscattering Born approximation for a biharmonic operator in 3D

We consider a backscattering Born approximation for a perturbed biharmonic operator in three space dimensions. Previous results on this approach for biharmonic operator use the fact that the coefficients are real-valued to obtain the reconstruction of singularities in the coefficients. In this text we drop the assumption about real-valued coefficients and also establish the recovery of singularities for complex coefficients. The proof uses mapping properties of the Radon transform.

Eigenvalues of the $p(x)-$biharmonic operator with indefinite weight] {Eigenvalues of the $p(x)-$biharmonic operator with indefinite weight under Neumann boundary conditions

In this paper we will study the existence of solutions for the nonhomogeneous elliptic equation with variable exponent $\Delta^2_{p(x)} u=\lambda V(x) |u|^{q(x)-2} u$, in a smooth bounded domain,under Neumann boundary conditions, where $\lambda$ is a positive real number, $p,q: \overline{\Omega} \rightarrow \mathbb{R}$, are continuous functions, and $V$ is an indefinite weight function. Considering different situations concerning the growth rates involved in the above quoted problem, we will prove the existence of a continuous family of eigenvalues.

Scattering problems for perturbations of the multidimensional biharmonic operator

Some scattering problems for the multidimensional biharmonic operator are studied. The operator is perturbed by first and zero order perturbations, which maybe complex-valued and singular. We show that the solutions to direct scattering problem satisfy a Lippmann-Schwinger equation, and that this integral equation has a unique solution in the weighted Sobolev space \begin{document}$H_{-δ}^2 $\end{document} . The main result of this paper is the proof of Saito's formula, which can be used to prove a uniqueness theorem for the inverse scattering problem. The proof of Saito's formula is based on norm estimates for the resolvent of the direct operator in \begin{document}$H_{-δ}^1 $\end{document} .

Nontrivial solutions of nonlocal fourth order elliptic equation of Kirchhoff type in [formula omitted

This paper is devoted to a class of important and general nonlocal fourth order elliptic problemΔ2u−(1+λ∫R3|∇u|2dx)Δu+V(x)u=f(x,u) in R3, where Δ2=Δ(Δ) is the bi-harmonic operator, λ≥0 is a constant. We focus on the case that f(x,u) involves a combination of convex and concave terms and the potential V(x) is allowed to be sign-changing. By new techniques, multiplicity results of two different type of solutions are established. Our results improves and generalizes that obtained in the literature.

Existence of solutions for a nonhomogeneous p(x)-biharmonic problem

This article is devoted to study the existence of solutions of nonlinear problem of fourth order governed by p(x)-biharmonic operator under certain conditions on f. Our technical approach is based on the Minty-Browder theorem applied to compact operators and operators of (S+) type.

Inverse backscattering problem for perturbations of biharmonic operator

We consider the inverse backscattering problem for a biharmonic operator with two lower order perturbations in two and three dimensions. The inverse Born approximation is used to recover jumps and singularities of an unknown combination of potentials. Numerical examples are given to illustrate the practical usefulness of the method.

The uniform convergence of the eigenfunctions expansions of the biharmonic operator in closed domain

The mathematical models of the various vibrating systems are partial differential equations and finding the solutions of such equations are obtained by developing the theory of eigenfunction expansions of differential operators. The biharmonic equation which is fourth order differential equation is encountered in plane problems of elasticity. It is also used to describe slow flows of viscous incompressible fluids. Many physical process taking place in real space can be described using the spectral theory of differentiable operators, particularly biharmonic operator. In this paper, the problems on the uniform convergence of eigenfunction expansions of the functions from Nikolskii classes corresponding to the biharmonic operator are investigated.

Spectral Analysis of the Biharmonic Operator Subject to Neumann Boundary Conditions on Dumbbell Domains

We consider the biharmonic operator subject to homogeneous boundary conditions of Neumann type on a planar dumbbell domain which consists of two disjoint domains connected by a thin channel. We analyse the spectral behaviour of the operator, characterizing the limit of the eigenvalues and of the eigenprojections as the thickness of the channel goes to zero. In applications to linear elasticity, the fourth order operator under consideration is related to the deformation of a free elastic plate, a part of which shrinks to a segment. In contrast to what happens with the classical second order case, it turns out that the limiting equation is here distorted by a strange factor depending on a parameter which plays the role of the Poisson coefficient of the represented plate.