Fourth-order Elliptic Operator Research Articles

Unique continuation and inverse problem for an anisotropic beam bending equation

This article studies the unique continuation of the principal and inverse problem of a perturbed anisotropic fourth-order elliptic operator arising in elastic beam theory, where the principle part of the operator is modelled by the non-rotational symmetric operator ∑j=1nDxj4. We discuss the three-ball inequality, stability estimate, strong unique continuation principle, and finally the inverse boundary value problem of recovering the coefficients featuring the bending stiffness of the material.

A path integral formula for certain fourth order elliptic operators

A path integral formula for certain fourth order elliptic operators

Equilibrium validation for triblock copolymers via inverse norm bounds for fourth-order elliptic operators

Block copolymers play an important role in materials sciences and have found widespread use in many applications. From a mathematical perspective, they are governed by a nonlinear fourth-order partial differential equation which is a suitable gradient of the Ohta–Kawasaki energy. While the equilibrium states associated with this equation are of central importance for the description of the dynamics of block copolymers, their mathematical study remains challenging. In the current paper, we develop computer-assisted proof methods which can be used to study equilibrium solutions in block copolymers consisting of more than two monomer chains, with a focus on triblock copolymers. This is achieved by establishing a computer-assisted proof technique for bounding the norm of the inverses of certain fourth-order elliptic operators, in combination with an application of a constructive version of the implicit function theorem. While these results are only applied to the triblock copolymer case, the obtained norm estimates can also be directly used in other contexts such as the rigorous verification of bifurcation points, or pseudo-arclength continuation in fourth-order parabolic problems.

Inequalities for eigenvalues of fourth-order elliptic operators in divergence form on complete Riemannian manifolds

We prove some inequalities of Payne–Pólya–Weinberger–Yang type for eigenvalues of fourth-order elliptic operators in weighted divergence form on complete Riemannian manifolds which generalizes the corresponding result for the clamped plate problem. We also prove estimates for lower-order eigenvalues that contain some of the estimates from the literature. As an application of our results, we obtain eigenvalues estimates for the bi-drifted Cheng–Yau operator.

A survey on obstacle-type problems for fourth order elliptic operators

A survey on obstacle-type problems for fourth order elliptic operators

Improved Approximations of Resolvents in Homogenization of Fourth Order Operators

We study homogenization of fourth order elliptic operators Ae in divergence form with e-periodic coefficients in ℝd and obtain an e2-order approximation of the resolvents (Ae + 1)−1 in the energy operator (L2→H2)-norm as e → 0.

Approximation of resolvents in homogenization of fourth-order elliptic operators

We study the homogenization of a fourth-order divergent elliptic operator with rapidly oscillating -periodic coefficients, where is a small parameter. The homogenized operator is of the same type and has constant coefficients. We apply Zhikov’s shift method to obtain an estimate in the -operator norm of order for the difference of the resolvents and . Bibliography: 25 titles.

Homogenization Estimates for Singularly Perturbed Operators

We study homogenization of a singularly perturbed second order elliptic operator Ae acting in the whole space ℝd with e-periodic coefficients as e → 0. Perturbation is generated by a fourth order elliptic operator, for example, by the bi-Laplacian Δ2 with a small parameter e2. We find approximations of the resolvent (Le + 1)−1 of order e and e2 in different operator norms. Bibliography: 28 titles.

Homogenization of the Fourth-Order Elliptic Operator with Periodic Coefficients with Correctors Taken into Account

An elliptic fourth-order differential operator $$A_\varepsilon$$ on $$L_2(\mathbb{R}^d;\mathbb{C}^n)$$ is studied. Here $$\varepsilon >0$$ is a small parameter. It is assumed that the operator is given in the factorized form $$A_\varepsilon = b(\mathbf{D})^* g(\mathbf{x}/\varepsilon) b(\mathbf{D})$$ , where $$g(\mathbf{x})$$ is a Hermitian matrix-valued function periodic with respect to some lattice and $$b(\mathbf{D})$$ is a matrix second-order differential operator. We make assumptions ensuring that the operator $$A_\varepsilon$$ is strongly elliptic. The following approximation for the resolvent $$(A_\varepsilon + I)^{-1}$$ in the operator norm of $$L_2(\mathbb{R}^d;\mathbb{C}^n)$$ is obtained: $$(A_{\varepsilon}+I)^{-1}=(A^{0}+I)^{-1}+\varepsilon K_{1}+\varepsilon^{2} K_{2}(\varepsilon)+O(\varepsilon^{3}).$$ Here $$A^0$$ is the effective operator with constant coefficients and $$K_{1}$$ and $$K_{2}(\varepsilon)$$ are certain correctors.

Homogenization of the Fourth-Order Elliptic Operator with Periodic Coefficients with Correctors Taken into Account

В $L_2(\mathbb{R}^d;\mathbb{C}^n)$ изучается эллиптический дифференциальный оператор $A_\varepsilon$ четвертого порядка. Здесь $\varepsilon >0$ - малый параметр. Предполагается, что оператор задан в факторизованном виде $A_\varepsilon = b(\mathbf{D})^* g(\mathbf{x}/\varepsilon)b(\mathbf{D})$, где эрмитова матрица-функция $g(\mathbf{x})$ периодична относительно некоторой решетки, а $b(\mathbf{D})$ - матричный дифференциальный оператор второго порядка. Делаются предположения, обеспечивающие сильную эллиптичность оператора $A_\varepsilon$. Получена аппроксимация резольвенты $(A_\varepsilon + I)^{-1}$ по операторной норме в $L_2(\mathbb{R}^d;\mathbb{C}^n)$ вида $$ (A_{\varepsilon}+I)^{-1}=(A^{0}+I)^{-1}+\varepsilon K_{1}+\varepsilon^{2} K_{2}(\varepsilon)+O(\varepsilon^{3}). $$ Здесь $A^0$ - эффективный оператор с постоянными коэффициентами, а $K_{1}$ и $K_{2}(\varepsilon)$ - некоторые корректоры.

Spectral inequality and resolvent estimate for the bi-Laplace operator

On a compact Riemannian manifold with boundary, we prove a spectral inequality for the bi-Laplace operator in the case of so-called clamped boundary conditions , that is, homogeneous Dirichlet and Neumann conditions simultaneously. We also prove a resolvent estimate for the generator of the damped plate semigroup associated with these boundary conditions. The spectral inequality allows one to observe finite sums of eigenfunctions for this fourth-order elliptic operator, from an arbitrary open subset of the manifold. Moreover, the constant that appears in the inequality grows as exp(Cµ 1/4) where µ is the largest eigenvalue associated with the eigenfunctions appearing in the sum. This type of inequality is known for the Laplace operator. As an application, we obtain a null-controllability result for a higher-order parabolic equation. The resolvent estimate provides the spectral behavior of the plate semigroup generator on the imaginary axis. This type of estimate is known in the case of the damped wave semigroup. As an application , we deduce a stabilization result for the damped plate equation, with a log-type decay. The proofs of both the spectral inequality and the resolvent estimate are based on the derivation of different types of Carleman estimates for an elliptic operator related to the bi-Laplace operator: in the interior and at some boundaries. One of these estimates exhibits a loss of one full derivative. Its proof requires the introduction of an appropriate semi-classical calculus and a delicate microlocal argument.

Inequalities for eigenvalues of fourth order elliptic operators in divergence form on Riemannian manifolds

In this paper, we study eigenvalue of linear fourth order elliptic operators in divergence form with Dirichlet boundary condition on a bounded domain in a compact Riemannian manifolds with boundary (possibly empty) and provide a general inequality for them. As an application, by using this inequality, we study eigenvalues of this operator on compact domains of complete submanifolds in a Euclidean space.

Exponentially Accurate Spectral Element Method for Fourth Order Elliptic Problems

In this paper, a fully non-conforming least-squares spectral element method for fourth order elliptic problems on smooth domains is presented. The proposed method works for a general fourth order elliptic operator with non-homogeneous data in two dimensions and gives exponentially accurate solutions. We derive differentiability estimates and prove our main stability estimate theorem using a non-conforming spectral element method. We then formulate a numerical scheme using a block diagonal preconditioner. Error estimates are also proven for the proposed method. We provide the computational complexity of our method and present results of numerical simulations that have been performed to validate the theory.

The Lebeau–Robbiano inequality for the one-dimensional fourth order elliptic operator and its application

In this paper, we establish the Lebeau–Robbiano inequality for the one-dimensional fourth order elliptic operator by using a point-wise estimate. Based on this inequality, we obtain the null controllability of one-dimensional stochastic fractional order Cahn–Hilliard equation.

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.

Vibrations of thin plates with small clamped patches

Eigenvalues of fourth-order elliptic operators feature prominently in stability analysis of elastic structures. This paper considers out-of-plane modes of vibration of a thin elastic plate perforated by a collection of small clamped patches. As the radius of each patch shrinks to zero, a point constraint eigenvalue problem is derived in which each patch is replaced by a homogeneous Dirichlet condition at its centre. The limiting problem is consequently not the eigenvalue problem with no patches, but a new type of spectral problem. The discrepancy between the eigenvalues of the patch-free and point constraint problems is calculated. The dependence of the point constraint eigenvalues on the location(s) of clamping is studied numerically using techniques from numerical algebraic geometry. The vibrational frequencies are found to depend very sensitively on the number and centre(s) of the clamped patches. For a range of number of punctures, we find spatial clamping patterns that correspond to local maxima of the base vibrational frequency of the plate.

Sign-preserving property for some fourth-order elliptic operators in one dimension or in radial symmetry

For a class of one-dimensional linear elliptic fourth-order equations with homogeneous Dirichlet boundary conditions it is shown that a non-positive and non-vanishing right-hand side gives rise to a negative solution. A similar result is obtained for the same class of equations for radially symmetric solutions in a ball or in an annulus. Several applications are given, including applications to nonlinear equations and eigenvalue problems.

Estimates for eigenvalues of fourth-order elliptic operators in divergence form

In this paper, we study the eigenvalue problem of fourth-order elliptic operators in divergence form with weight on compact Riemannian manifolds with boundary (possibly empty) and get two types of general inequalities for them. By using these general inequalities, we can obtain some universal inequalities for eigenvalues of fourth-order elliptic operators in divergence form with weight on bounded domains in complete Riemannian manifolds. Furthermore, we also obtain some estimates for lower order eigenvalues of fourth-order elliptic operators in divergence form with weight on compact submanifolds in Euclidean spaces.

Eventually positive semigroups of linear operators

We develop a systematic theory of eventually positive semigroups of linear operators mainly on spaces of continuous functions. By eventually positive we mean that for every positive initial condition the solution to the corresponding Cauchy problem is positive for large enough time. Characterisations of such semigroups are given by means of resolvent properties of the generator and Perron–Frobenius type spectral conditions. We apply these characterisations to prove eventual positivity of several examples of semigroups including some generated by fourth order elliptic operators and a delay differential equation. We also consider eventually positive semigroups on arbitrary Banach lattices and establish several results for their spectral bound which were previously only known for positive semigroups.

Spectral asymptotics for a singularly perturbed fourth order locally periodic elliptic operator

We consider the homogenization of a singularly perturbed self-adjoint fourth order elliptic operator with locally periodic coefficients, stated in a bounded domain. We impose Dirichlet boundary conditions on the boundary of the domain. The presence of large parameters in the lower order terms and t he dependence of the coefficients on the slow variable lead to localization of the eigenfunctions. We show that the jth eigenfunction can be approximated by a rescaled function that is constructed in terms of the jth eigenfunction of fourth or second order effective operators with constant coefficients.