Fourth-order Elliptic Problems Research Articles

Positive solutions to a fourth-order elliptic problem by the Lusternik–Schnirelmann category

In this paper we consider the fourth-order problem{Δ2u=μ|u|s−1u+|u|2⁎−2uinΩ,u,−Δu>0in Ω,u,Δu=0on ∂Ω, where Ω is a smooth bounded domain in RN, N≥5 and 2⁎=2N/(N−4). We assume 2≤s+1<2⁎ in case N≥8 and 2⁎−2<s+1<2⁎ for the critical dimensions N=5,6,7. Then we prove that if Ω has a rich topology, described by its Lusternik–Schnirelmann category, then the problem has multiple solutions, at least as many as catΩ(Ω), in case the parameter μ>0 is sufficiently small.

Uniformly Convergent Nonconforming Element for 3-D Fourth Order Elliptic Singular Perturbation Problem

In this paper, using a bubble function, we construct a cuboid element to solve the fourth order elliptic singular perturbation problem in three dimensions. We prove that the nonconforming C 0 -cuboid element converges in the energy norm uniformly with respect to the perturbation parameter. Mathematics subject classification: 65N12, 65N30.

Numerical Computation of Stationary Solitary Waves to 2D Boussinesq Equation

The aim of this work is to evaluate stationary moving solitary wave solutions to the 2D Boussinesq equation. To solve the resulting nonlinear fourth order elliptic problem we use a combination of high order finite difference schemes, an iterative procedure and new asymptotic boundary conditions. Numerical tests with several finite difference schemes (of 2nd, 4th, and 6th order of approximation), a variety of parameters of the problem and two initial approximations for the iterative process (the best fit formulas given in in the paper C. I. Christov, J. Choudhury, Mech. Res. Commun., 38 (2011) 274 - 281 and the solution to the ground state equation) show that the numerical solutions converge with a high accuracy to the one and the same solution of the initial problem.

A posteriori error estimates of the Morley element for the fourth order elliptic eigenvalue problem

This paper is devoted to a posteriori error estimates of the Morley element for the fourth order elliptic eigenvalue problem. The reliability and the efficiency of the a posteriori error estimates are proved in the paper. Numerical examples illustrate the advantages of the adaptive Morley element method for the fourth order elliptic eigenvalue problem.

Global homeomorphism and applications to the existence and uniqueness of solutions of some differential equations

In this paper, the global diffeomorphism theorem due to Zampieri was generalized. The concept of the basin of attraction is the main tool of our exposition in discussing the diffeomorphism between Banach spaces. The existence and uniqueness of a solution of the fourth order elliptic boundary value problem was proved by employing our generalized theorem. The results of this paper generalize some known theorems.MSC:35J40.

Multiple solutions for a fourth-order nonlinear elliptic problem which is superlinear at +∞ and linear at −∞

We consider a semilinear fourth-order elliptic equation with a right-hand side nonlinearity which exhibits an asymmetric growth at +∞ and at −∞. Namely, it is linear at −∞ and superlinear at +∞. Combining variational methods with Morse theory, we show that the problem has at least two nontrivial solutions, one of which is negative.

Existence of Multiple Solutions for Fourth-Order Elliptic Problem

By using the variant fountain theorem, we study the existence of multiple solutions for a class of superquadratic fourth-order elliptic problem with Navier boundary value condition.

Convergence Analysis of Incomplete Biquadratic Rectangular Element for Fourth-Order Singular Perturbation Problem on Anisotropic Meshes

The convergence analysis of a Morley type rectangular element for the fourth-order elliptic singular perturbation problem is considered. A counterexample is provided to show that the element is not uniformly convergent with respect to the perturbation parameter. A modified finite element approximation scheme is used to get convergent results; the corresponding error estimate is presented under anisotropic meshes. Numerical experiments are also carried out to demonstrate the theoretical analysis.

Fourth order elliptic boundary value problem with nonlinear term decaying at the origin

We consider the number of the weak solutions for some fourth order elliptic boundary value problem with bounded nonlinear term decaying at the origin. We get a theorem, which shows the existence of the bounded solution for this problem. We obtain this result by approaching the variational method and using the generalized mountain pass theorem for the fourth order elliptic problem with bounded nonlinear term.MSC:35J30, 35J40.

On a fourth order elliptic problem with ap(x)-biharmonic operator

Sufficient conditions are obtained for the existence of at least one nontrivial weak solution to a fourth order elliptic problem with a p ( x ) -biharmonic operator and the Navier boundary conditions.

A Petrov-Galerkin spectral method for fourth-order problems

In this paper, we consider the Petrov–Galerkin spectral method for fourth-order elliptic problems on rectangular domains subject to non-homogeneous Dirichlet boundary conditions. We derive some sharp results on the orthogonal approximations in one and two dimensions, which play important roles in numerical solutions of higher-order problems. By applying these results to a fourth-order problem, we establish the H2-error and L2-error bounds of the Petrov–Galerkin spectral method. Numerical experiments are provided to illustrate the high accuracy of the proposed method and coincide well with the theoretical analysis. Copyright © 2013 John Wiley & Sons, Ltd.

Numerical controllability of the wave equation through primal methods and Carleman estimates

This paper deals with the numerical computation of boundary null controls for the 1D wave equation with a potential. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. We do not apply in this work the usual duality arguments but explore instead a direct approach in the framework of global Carleman estimates. More precisely, we consider the control that minimizes over the class of admissible null controls a functional involving weighted integrals of the state and the control. The optimality conditions show that both the optimal control and the associated state are expressed in terms of a new variable, the solution of a fourth-order elliptic problem defined in the space-time domain. We first prove that, for some specific weights determined by the global Carleman inequalities for the wave equation, this problem is well-posed. Then, in the framework of the finite element method, we introduce a family of finite-dimensional approximate control problems and we prove a strong convergence result. Numerical experiments confirm the analysis. We complete our study with several comments.

ERROR ESTIMATES OF MIXED FINITE ELEMENT APPROXIMATIONS FOR A CLASS OF FOURTH ORDER ELLIPTIC CONTROL PROBLEMS

In this paper, we consider the error estimates of the numerical solutions of a class of fourth order linear-quadratic elliptic optimal control problems by using mixed finite element methods. The state and co-state are approximated by the order k Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise polynomials of order k (k � 1). L 2 and L ∞ -error estimates are derived for both the control and the state approximations. These results are seemed to be new in the literature of the mixed finite element methods for fourth order elliptic control problems.

On the perturbed Q-curvature problem on [formula omitted

Let g0 denote the standard metric on S4 and Pg0=Δg02−2Δg0 denote the corresponding Paneitz operator. In this work, we study the following fourth order elliptic problem with exponential nonlinearityPg0u+6=2Q(x)e4uon S4. Here Q is a prescribed smooth function on S4 which is assumed to be a perturbation of a constant. We prove existence results to the above problem under assumptions only on the “shape” of Q near its critical points. These are more general than the non-degeneracy conditions assumed so far. We also show local uniqueness and exact multiplicity results for this problem. The main tool used is the Lyapunov–Schmidt reduction.

$$C^{0}$$ -nonconforming tetrahedral and cuboid elements for the three-dimensional fourth order elliptic problem

In this paper, a theoretical framework is constructed on how to develop $$C^0$$ -nonconforming elements for the fourth order elliptic problem. By using the bubble functions, a simple practical method is presented to construct one tetrahedral $$C^{0}$$ -nonconforming element and two cuboid $$C^{0}$$ -nonconforming elements for the fourth order elliptic problem in three spacial dimensions. It is also proved that one element is of first order convergence and other two are of second order convergence. From the best knowledge of us, this is the first success in constructing the second-order convergent nonconforming element for the fourth order elliptic problem.

C 0-Nonconforming Triangular Prism Elements for the Three-Dimensional Fourth Order Elliptic Problem

In this paper, using the bubble functions, we construct two C 0-nonconforming triangular prism elements for the fourth order elliptic problem in three dimensions. By the abstract convergence theorem in (Chen et al. in Numer. Math. (2012, accepted)), one element is proved to be of first order convergence and the other one is proved to be of second order convergence.

Properties of the extremal solution for a fourth-order elliptic problem

Let λ* &gt; 0 denote the largest possible value of λ such that the systemhas a solution, where $\mathbb{B}$ is the unit ball in ℝn centred at the origin, p &gt; 1 and n is the exterior unit normal vector. We show that for λ = λ* this problem possesses a unique weak solution u*, called the extremal solution. We prove that u* is singular when n ≥ 13 for p large enough and actually solve part of the open problem which Dávila et al. left unsolved.

Uniformly stable rectangular elements for fourth order elliptic singular perturbation problems

AbstractIn this article, we consider rectangular finite element methods for fourth order elliptic singular perturbation problems. We show that the non‐ C0 rectangular Morley element is uniformly convergent in the energy norm with respect to the perturbation parameter. We also propose a C0 extended high order rectangular Morley element and prove the uniform convergence. Finally, we do some numerical experiments to confirm the theoretical results. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

Multiple solutions for a fourth-order nonlinear elliptic problem

The existence of multiple solutions for a class of fourth-order elliptic equation with respect to the generalized asymptotically linear conditions is established by using the minimax method and Morse theory.

ELLIPTIC PROBLEM WITH A VARIABLE COEFFICIENT AND A JUMPING SEMILINEAR TERM

We obtain the multiple solutions for the fourth order elliptic problem with a variable coefficient and a jumping semilinear term. We have a result that there exist at least two solutions if the variable coefficient of the semilinear term crosses some number of the eigenvalues of the biharmonic eigenvalue problem. We obtain this multiplicity result by applying the Leray-Schauder degree theory.