Biharmonic Operator Research Articles

Ground states for a class of asymptotically linear fourth-order elliptic equations

In this paper, we study the following fourth-order elliptic equation,where is the biharmonic operator, and . We apply the variational methods to obtain the existence of ground-state solutions for asymptotically linear case.

Existence and multiplicity of nontrivial solutions for a class of modified nonlinear fourth-order elliptic equations on [formula omitted

In this paper, we study the following fourth-order elliptic equation of the form▵2u-▵u+V(x)u-κ2▵(u2)u=h(x,u),x∈RN,where ▵2≔▵(▵) is the biharmonic operator, κ⩾0, N⩽6, V∈C(RN,R) and h∈C(RN×R,R). Under appropriate assumptions on V(x) and h(x,u), some existence and multiplicity results for nontrivial solutions are obtained via variational methods.

Global existence versus blow-up results for a fourth order parabolic PDE involving the Hessian

We consider a partial differential equation that arises in the coarse-grained description of epitaxial growth processes. This is a parabolic equation whose evolution is governed by the competition between the determinant of the Hessian matrix of the solution and the biharmonic operator. This model might present a gradient flow structure depending on the boundary conditions. We first extend previous results on the existence of stationary solutions to this model for Dirichlet boundary conditions. For the evolution problem we prove local existence of solutions for arbitrary data and global existence of solutions for small data. By exploiting the boundary conditions and the variational structure of the equation, according to the size of the data we prove finite time blow-up of the solution and/or convergence to a stationary solution for global solutions.

An interior penalty method for distributed optimal control problems governed by the biharmonic operator

In this paper, a C0 interior penalty method has been proposed and analyzed for distributed optimal control problems governed by the biharmonic operator. The state and adjoint variables are discretized using continuous piecewise quadratic finite elements while the control variable is discretized using piecewise constant approximations. A priori and a posteriori error estimates are derived for the state, adjoint and control variables under minimal regularity assumptions. Numerical results justify the theoretical results obtained. The a posteriori error estimators are useful in adaptive finite element approximation and the numerical results indicate that the sharp error estimators work efficiently in guiding the mesh refinement.

Eigenvalues for a fourth order elliptic problem

We study the fourth order nonlinear eigenvalue problem with a $p(x)$-biharmonic operator \begin{equation*} \left \{\begin {array}{l} \Delta ^2_{p(x)}u+a(x)|u|^{p(x)-2}u=\lambda w(x)f(u)\quad \text {in}\ \Omega , \\ u=\Delta u=0\quad \text {on}\ \partial \Omega , \end{array} \right . \end{equation*} where $\Omega$ is a smooth bounded domain in $\mathbb {R}^N$, $p\in C(\overline {\Omega })$ with $p(x)>1$ on $\overline {\Omega }$, $\Delta ^2_{p(x)}u=\Delta (|\Delta u|^{p(x)-2}\Delta u)$ is the $p(x)$-biharmonic operator, and $\lambda >0$ is a parameter. Under some appropriate conditions on the functions $p, a, w, f$, we prove that there exists $\overline {\lambda }>0$ such that any $\lambda \in (0,\overline {\lambda })$ is an eigenvalue of the above problem. Our analysis mainly relies on variational arguments based on Ekeland’s variational principle and some recent theory on the generalized Lebesgue–Sobolev spaces $L^{p(x)}(\Omega )$ and $W^{k,p(x)}(\Omega )$.

Determining the first order perturbation of a bi-harmonic operator on bounded and unbounded domains from partial data

In this paper we study inverse boundary value problems with partial data for the bi-harmonic operator with first order perturbation. We consider two types of subsets of Rn(n≥3), one is an infinite slab, the other is a bounded domain. In the case of a slab, we show that, from Dirichlet and Neumann data given either on the different boundary hyperplanes of the slab or on the same boundary hyperplane, one can uniquely determine the magnetic potential and the electric potential.In the case of a bounded domain, we show the unique determination of the magnetic potential and the electric potential from partial Dirichlet and Neumann data under two different assumptions. The first assumption is that the magnetic and electric potentials are known in a neighborhood of the boundary, in this situation we obtain the uniqueness result when the Dirichlet and Neumann data are only given on two arbitrary open subsets of the boundary. The second assumption is that the Dirichlet and Neumann data are known on the same part of the boundary whose complement is a part of a hyperplane, we also establish the unique determination result in this local data case.

Existence and Multiplicity of Solutions for Kirchhoff Type Problems Involving $p(x)$-Biharmonic Operators

This paper is concerned with the existence and multiplicity of weak solutions for a p(x) -Kirchhoff type problem of the following form \left\{ \begin{alignedat}{2} M\left(\int_{\Omega}\frac{1}{p(x)}|\Delta u|^{p(x)}\,dx\right)\Delta(|\Delta u|^{p(x)-2}\Delta u) &=f(x,u)& &\text{ in } \Omega \\ u&=\Delta u =0 &\quad &\textrm{ on } \partial\Omega, \end{alignedat}\right. by using the mountain pass theorem of Ambrosetti and Rabinowitz and Ekeland's variational principle in two cases when the Carathéodory function f(x,u) having special structure.

Nonlinear Biharmonic Problems with Singular Potentials

We deal with the problem\begin{eqnarray}\Delta^2 u +V(|x|)u = f(u), u\in D^{2,2}(R^N)\end{eqnarray}where $\Delta^2$ is biharmonic operator and the potential $V > 0 $ is measurable, singular at the origin and may also have a continuousset of singularities. The nonlinearity is continuous and has a super-linear power-like behaviour;both sub-critical and super-critical cases are considered. We prove the existence of nontrivial radialsolutions. If $f$ is odd, we show that the problem has infinitely many radial solutions.

Hardy–Sobolev inequalities for the biharmonic operator with remainder terms

We first review improvements of (first-order) Sobolev and Hardy inequalities by the addition of suitable lower-order terms (these improved inequalities have been pioneered by Brezis and Nirenberg (1983) and Brezis and Vazquez (1997)). Recently, corresponding results concerning first-order Hardy–Sobolev and higher-order Sobolev and Hardy inequalities have been proved.

Existence of weak solutions for a fourth-order Navier boundary value problem

In this paper we discuss the existence of weak solutions for the fourth-order Navier boundary value problem { Δ 2 u ( x ) + c Δ u ( x ) = λ u ( x ) + f ( u ( x ) ) , in Ω , u = Δ u = 0 , on ∂ Ω , where λ is a parameter, Δ 2 is the biharmonic operator, Ω ⊂ R N ( N > 4 ) is a smooth bounded domain, and f ∈ C ( R , R ) . We use topological degree theory and critical point theory to establish the existence.

Platonic scattering cancellation for bending waves in a thin plate.

We propose an ultra-thin elastic cloak to control the scattering of bending waves in isotropic heterogeneous thin plates. The cloak design makes use of the scattering cancellation technique applied, for the first time, to the biharmonic operator describing the propagation of bending waves in thin plates. We first analyze scattering from hard and soft cylindrical objects in the quasistatic limit, then we prove that the scattering of bending waves from an object in the near and far-field regions can be suppressed significantly by covering it with a suitably designed coating. Beyond camouflaging, these findings may have potential applications in protection of buildings from earthquakes and isolating structures from vibrations in the motor vehicle industry.

Multiple Results to Some Biharmonic Problems

We study a nonlinear elliptic problem defined in a bounded domain involving biharmonic operator together with an asymptotically linear term. We establish at least three nontrivial solutions using the topological degree theory and the critical groups.

Boundary eigencurve problems involving the biharmonic operator

Boundary eigencurve problems involving the biharmonic operator

EXISTENCE OF THREE SOLUTIONS FOR A NAVIER BOUNDARY VALUE PROBLEM INVOLVING THE p(x)-BIHARMONIC

The existence of at least three weak solutions is established for a class of quasilinear elliptic equations involving the p(x)-biharmonic operators with Navier boundary value conditions. The technical approach is mainly based on a three critical points theorem due to Ricceri [11].

Asymptotics for the principal eigenvalue of the [formula omitted]-biharmonic operator on the ball as [formula omitted] approaches 1

Using the estimates of the principal eigenvalue of the p-biharmonic operator on the ball from below and from above, we provide its asymptotic analysis as p→1+.

Separation of bi-harmonic differential operators on Riemannian manifolds

Abstract. Consider the bi-harmonic differential expression of the form A = ▵ ▵ + q ${A=\triangle \triangle +q}$ on a complete Riemannian manifold (M,g) with metric g , ${g,}$ where ▵ ${\triangle }$ is the Laplacian on M and q ≥ 0 ${q\ge 0}$ is a locally square integrable function on M. In the terminology of Everitt and Giertz, the differential expression A is said to be separated in L 2 ( M ) ${L^{2}( M) }$ if for all u ∈ L 2 ( M ) ${u\in L^{2}( M) }$ such that A u ∈ L 2 ( M ) ${Au\in L^{2}( M) }$ , we have q u ∈ L 2 ( M ) ${ qu\in L^{2}( M)}$ . In this paper we give sufficient conditions for A to be separated in L 2 ( M ) ${L^{2}( M)}$ .

Existence result for a class of biharmonic equations with critical growth and singular potential inRN

In this paper, we prove a compactness theorem for a critical Sobolev embedding involving the biharmonic operator with a singular potential. Then, using the compactness theorem and the Mountain Pass Theorem, we obtain the existence of nontrivial solutions for a class of biharmonic equations with critical growth and a singular potential in R N .

Sharp spectral multipliers for operators satisfying generalized Gaussian estimates

Let L be a non-negative self-adjoint operator acting on L2(X) where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e−tL which satisfies generalized m-th order Gaussian estimates. In this article, we study singular and dyadically supported spectral multipliers for abstract self-adjoint operators. We show that in this setting sharp spectral multiplier results follow from Plancherel or Stein–Tomas type estimates. These results are applicable to spectral multipliers for a large class of operators including m-th order elliptic differential operators with constant coefficients, biharmonic operators with rough potentials and Laplace type operators acting on fractals.

Spectral stability results for higher-order operators under perturbations of the domain

We analyze the spectral behavior of higher-order elliptic operators when the domain is perturbed. We provide general spectral stability results for Dirichlet and Neumann boundary conditions. Moreover, we study the bi-harmonic operator with the so-called intermediate boundary conditions. We give special attention to this last case and analyze its behavior when the boundary of the domain has some oscillatory behavior. We will show that there is a critical oscillatory behavior and that the limit problem depends on whether we are above, below or just sitting on this critical value.

MULTIPLICITY OF SOLUTIONS FOR BIHARMONIC ELLIPTIC SYSTEMS INVOLVING CRITICAL NONLINEARITY

In this paper, we consider the biharmonic elliptic systems of the form <TEX>$$\{{\Delta}^2u=F_u(u,v)+{\lambda}{\mid}u{\mid}^{q-2}u,\;x{\in}{\Omega},\\{\Delta}^2v=F_v(u,v)+{\delta}{\mid}v{\mid}^{q-2}v,\;x{\in}{\Omega},\\u=\frac{{\partial}u}{{\partial}n}=0,\; v=\frac{{\partial}v}{{\partial}n}=0,\;x{\in}{\partial}{\Omega},$$</TEX>, where <TEX>${\Omega}{\subset}\mathbb{R}^N$</TEX> is a bounded domain with smooth boundary <TEX>${\partial}{\Omega}$</TEX>, <TEX>${\Delta}^2$</TEX> is the biharmonic operator, <TEX>$N{\geq}5$</TEX>, <TEX>$2{\leq}q$</TEX> < <TEX>$2^*$</TEX>, <TEX>$2^*=\frac{2N}{N-4}$</TEX> denotes the critical Sobolev exponent, <TEX>$F{\in}C^1(\mathbb{R}^2,\mathbb{R}^+)$</TEX> is homogeneous function of degree <TEX>$2^*$</TEX>. By using the variational methods and the Ljusternik-Schnirelmann theory, we obtain multiplicity result of nontrivial solutions under certain hypotheses on <TEX>${\lambda}$</TEX> and <TEX>${\delta}$</TEX>.