Biharmonic Operator Research Articles

A Nonlocal Biharmonic Operator and its Connection with the Classical Analogue

We introduce here a nonlocal operator as a natural generalization to the biharmonic operator that appears in plate theory. This operator is built in the nonlocal calculus framework defined by Du et al. and its connected with the recent theory of peridynamics. For the steady state equation coupled with different boundary conditions we show existence and uniqueness of solutions, as well as regularity of solutions. The boundary conditions considered are nonlocal counterparts of the classical clamped and hinged boundary conditions. For each system we show convergence of the nonlocal solutions to their local equivalents using compactness arguments developed by Bourgain, Brezis and Mironescu.

Stability of the inverse boundary value problem for the biharmonic operator: Logarithmic estimates

Abstract In this article, we establish logarithmic stability estimates for the determination of the perturbation of the biharmonic operator from partial data measurements when the inaccessible part of the domain is flat and homogeneous boundary conditions are assumed on this part. This is an improvement to a log-log type stability estimate proved earlier for the partial data case.

Existence of multi-bump solutions for a class of elliptic problems involving the biharmonic operator

Using variational methods, we establish existence of multi-bump solutions for the following class of problems $$\begin{aligned} \left\{ \begin{array}{l} \Delta ^2 u +(\lambda V(x)+1)u = f(u), \quad \text{ in } \quad \mathbb {R}^{N}, u \in H^{2}(\mathbb {R}^{N}), \end{array} \right. \end{aligned}$$ where \(N \ge 1\), \(\Delta ^2\) is the biharmonic operator, f is a continuous function with subcritical growth, \(V : \mathbb {R}^N \rightarrow \mathbb {R}\) is a continuous function verifying some conditions and \(\lambda >0\) is a real constant large enough.

Magnetic Bi-harmonic differential operators on Riemannian manifolds and the separation problem

In this paper we obtain sufficient conditions for the bi-harmonic differential operator A = ΔE2 + q to be separated in the space L2 (M) on a complete Riemannian manifold (M,g) with metric g, where ΔE is the magnetic Laplacian onM and q ≥ 0 is a locally square integrable function on M. Recall that, in the terminology of Everitt and Giertz, the differential operator A is said to be separated in L2 (M) if for all u ∈ L2 (M) such that Au ∈ L2 (M) we have ΔE2u ∈ L2 (M) and qu ∈ L2 (M).

Weighted Hardy–Littlewood–Sobolev inequalities on the upper half space

In this paper, we establish a weighted Hardy–Littlewood–Sobolev (HLS) inequality on the upper half space using a weighted Hardy type inequality on the upper half space with boundary term, and discuss the existence of extremal functions based on symmetrization argument. As an application, we can show a weighted Sobolev–Hardy trace inequality with [Formula: see text]-biharmonic operator.

A note on the Neumann eigenvalues of the biharmonic operator

We study the dependence of the eigenvalues of the biharmonic operator subject to Neumann boundary conditions on the Poisson's ratio σ. In particular, we prove that the Neumann eigenvalues are Lipschitz continuous with respect to σ∈[0,1[and that all the Neumann eigenvalues tend to zero as σ→1−. Moreover, we show that the Neumann problem defined by setting σ = 1 admits a sequence of positive eigenvalues of finite multiplicity that are not limiting points for the Neumann eigenvalues with σ∈[0,1[as σ→1− and that coincide with the Dirichlet eigenvalues of the biharmonic operator. Copyright © 2016 John Wiley & Sons, Ltd.

Fourth-order Schrödinger type operator with singular potentials

In this paper we study the biharmonic operator perturbed by an inverse fourth-order potential. In particular, we consider the operator $A=\Delta^2-V=\Delta^2-c|x|^{-4}$ where $c$ is any constant such that $c<\left(\frac{N(N-4)}{4}\right)^2$. The semigroup generated by $-A$ in $L^2(\mathbb{R}^N)$, $N\geq5$, extrapolates to a bounded holomorphic $C_0$-semigroup on $L^p(\mathbb{R}^N)$ for $p\in [p^{'}_0,p_0]$ where $p_0=\frac{2N}{N-4}$ and $p_0^{'}$ is its dual exponent. Furthermore, we study the boundedness of the Riesz transform $\Delta A^{-1/2}$ on $L^p(\mathbb{R}^N)$ for all $p\in(p_0^{'},2]$.

Ground state solutions and least energy sign-changing solutions for a class of fourth order Kirchhoff-type equations in [formula omitted

In this article we study the problem Δ2u−(1+λ∫RN|∇u|2dx)Δu+V(x)u=|u|p−2uin RN, where Δ2≔Δ(Δ) is the biharmonic operator, λ>0 is a parameter, p∈(2,2∗), and V(x)∈C(RN,R). Under appropriate assumptions on V(x), the existence of ground state solutions and a least energy sign-changing solution is obtained by combining the variational methods and the Nehari method.

Existence of one weak solution for p(x)-biharmonic equations with Navier boundary conditions

We study the existence of at least one weak solution for a class of elliptic Navier boundary value problems involving the p(x)-biharmonic operator. Our technical approach is based on variational methods. In addition, an example to illustrate our results is given.

Picone’s identity for biharmonic operators on Heisenberg group and its applications

In this paper, we establish a nonlinear analogue of Picone’s identity for biharmonic operators on Heisenberg group. As an applications of Picone’s identity, we obtain Hardy-Rellich type inequality, Morse index, Caccioppoli inequality, Picone inequality for biharmonic operators on Heisenberg group.

Borg–Levinson theorem for perturbations of the bi-harmonic operator

We prove that the boundary spectral data, i.e. the Dirichlet eigenvalues and special derivatives up to the third order of the normalized eigenfunctions at the boundary uniquely determine the coefficients of the second order perturbation of the bi-harmonic operator in the smooth bounded domain .

A Note on Lower Bounds of Eigenvalues of Fourth Order Elliptic Operators on Local Quasi-Uniform Grids

Abstract This paper is a generalization and improvement of some recent results concerned with the lower bound property of eigenvalues produced by the Morley element of the biharmonic operator. Such an extension is of two fold. First, we generalize those results for the biharmonic operator to general fourth order elliptic operators; second, we extend them from quasi-uniform grids to local quasi-uniform grids. In particular, for the general case, we prove the asymptotic lower bound property of the discrete eigenvalues under a very weak saturation condition in terms of local meshsizes which can in general not be improved; for the special case with B = 0, we show a similar result without any assumption except the fineness of the mesh. These results can be regarded as a substantial improvement of a related result on regular meshes, including adaptive local refined meshes, for general fourth order elliptic operators, due to Yang, Li and Bi [http://arxiv.org/abs/1509.00566], while the analysis herein is completely different and largely simpler than that in the above paper.

Particular solutions of Laplace and bi-harmonic operators using Matérn radial basis functions

ABSTRACTIn this paper, we derive closed-form particular solutions of Matérn radial basis functions for the Laplace and biharmonic operator in 2D and Laplace operator in 3D. These derived particular solutions are essential for the implementation of the method of particular solutions for solving various types of partial differential equations. Four numerical examples in 2D and 3D are given to demonstrate the effectiveness of the derived particular solutions.

The lower bounds of the first eigenvalues for the biharmonic operator on manifolds

In this paper, we use the Reilly formula and the Hessian comparison theorem to estimate the lower bounds of the first eigenvalues for the biharmonic operator eigenvalue problems (buckling problem and clamped plate problem) on manifolds.

Multiple solutions for fourth order elliptic problems with p(x)-biharmonic operators

We study the multiplicity of weak solutions to the following fourth order nonlinear elliptic problem with a \(p(x)\)-biharmonic operator \[\begin{cases}\Delta^2_{p(x)}u+a(x)|u|^{p(x)-2}u=\lambda f(x,u)\quad\text{ in }\Omega, u=\Delta u=0\quad\text{ on }\partial\Omega,\end{cases}\] where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\), \(p\in C(\overline{\Omega})\), \(\Delta^2_{p(x)}u=\Delta(|\Delta u|^{p(x)-2}\Delta u)\) is the \(p(x)\)-biharmonic operator, and \(\lambda\gt 0\) is a parameter. We establish sufficient conditions under which there exists a positive number \(\lambda^{*}\) such that the above problem has at least two nontrivial weak solutions for each \(\lambda\gt\lambda^{*}\). Our analysis mainly relies on variational arguments based on the mountain pass lemma and some recent theory on the generalized Lebesgue-Sobolev spaces \(L^{p(x)}(\Omega)\) and \(W^{k,p(x)}(\Omega)\).

A Uniform Coerciveness Result for Biharmonic Operator and its Application to a Parabolic Equation

A Uniform Coerciveness Result for Biharmonic Operator and its Application to a Parabolic Equation

Nodal ground state solution to a biharmonic equation via dual method

Using the dual method we establish the existence of a nodal ground state solution for the following class of problems{Δ2u=f(u),inΩ,u=Bu=0,on∂Ω, where Ω⊂RN(N≥1) is a smooth bounded domain, Δ2 is the biharmonic operator, B=Δ or B=∂∂ν and f is a C1-function having subcritical growth.

REMARKS ON THE QUALITATIVE QUESTIONS FOR BIHARMONIC OPERATORS

In this article, we obtain several remarks on the qualitative questions such as Picone's identity, Morse index and Hardy-Rellich type inequality for biharmonic operators.

Boundary element method based vibration analysis of elastic bottom plates of fluid storage tanks resting on Pasternak foundation

A higher-order boundary element procedure is presented for the free vibration analysis of flexible base plates of rigid fluid storage tanks resting on elastic foundation. The main principles of the procedure are replacing the biharmonic operator of the thin plate vibration problem by two successive harmonic operators, and representing the forcing terms (plate inertia, fluid loading and foundation influence) by applying the dual reciprocity boundary element formulation. The fluid effect on the plate dynamics is incorporated into the analysis by invoking another boundary element solution, which expresses the fluid pressure over the plate surface in terms of plate deflection. The performance of the method is thoroughly investigated and the nature of dynamic plate–foundation–fluid interaction is studied from several perspectives. The method provides excellent predictions, within the limits of Kirchhoff plate, potential flow, and Pasternak foundation models.

Semilinear Elliptic Equation with Biharmonic Operator and Multiple Critical Nonlinearities

Abstract We study the existence and non-existence of nontrivial weak solution of , where N ≥ 5, and . Using Pohozaev type of identity, we prove the non-existence result when 1 &lt; q &lt; 2**. On the other hand when the equation has multiple critical nonlinearities i.e. q = 2** and −(N − 2)2 ≤ μ &lt; μ1, we establish the existence of nontrivial solution, using the Mountain-Pass theorem by Ambrosetti and Rabinowitz.