Fourth-order Elliptic Equations Research Articles

The Kirchhoff plate equation on surfaces: the surface Hellan–Herrmann–Johnson method

Abstract We present a mixed finite element method for approximating a fourth-order elliptic partial differential equation (PDE), the Kirchhoff plate equation, on a surface embedded in ${\mathbb {R}}^{3}$, with or without boundary. Error estimates are given in mesh-dependent norms that account for the surface approximation and the approximation of the surface PDE. The method is built on the classic Hellan–Herrmann–Johnson method (for flat domains), and convergence is established for $C^{k+1}$ surfaces, with degree $k$ (Lagrangian, parametrically curved) approximation of the surface, for any $k \geqslant 1$. Mixed boundary conditions are allowed, including clamped, simply-supported and free conditions; if free conditions are present then the surface must be at least $C^{2,1}$. The framework uses tools from differential geometry and is directly related to the seminal work of Dziuk, G. (1988) Finite elements for the Beltrami operator on arbitrary surfaces. Partial Differential Equations and Calculus of Variations, vol. 1357 (S. Hildebrandt & R. Leis eds). Berlin, Heidelberg: Springer, pp. 142–155. for approximating the Laplace–Beltrami equation. The analysis here is the first to handle the full surface Hessian operator directly. Numerical examples are given on nontrivial surfaces that demonstrate our convergence estimates. In addition, we show how the surface biharmonic equation can be solved with this method.

Multiple solutions for a fourth-order elliptic equation involving singularity

Here, we consider a fourth-order elliptic problem involving singularity and p(x)- biharmonic operator. Using Hardy’s inequality, S+-condition, and Palais-Smale condition, the existence of weak solutions in a bounded domain in RN is proved. Finally, we percent some examples.

Existence of nontrivial solutions for modified nonlinear fourth-order elliptic equations with indefinite potential

In this paper, we investigate the following modified nonlinear fourth-order elliptic equations{Δ2u−Δu+V(x)u−12uΔ(u2)=g(u),inRN,u∈H2(RN) where Δ2=Δ(Δ) is the biharmonic operator, V is an indefinite potential, g grows subcritically and satisfies the Ambrosetti-Rabinowitz type condition g(t)t≥μG(t)≥0 with μ>3. Using Morse theory, we obtain nontrivial solutions of the above equations. Our result complements recent results in [17], where g has to be 3-superlinear at infinity.

An isoparametric mixed finite element method for approximating a class of fourth-order elliptic problem

A Ciarlet-Raviart type isoparametric mixed finite element method (MFEM) is constructed and analyzed for solving a class of fourth-order elliptic equation with Navier boundary condition defined on a curved domain in R2, and numerical quadrature is also considered in the scheme. The existence and uniqueness of the numerical solutions are proved under certain numerical quadrature. With the help of the special technically analysis, the optimal error estimates with H1 norm are obtained in Ωh, which yields better accuracy than using a convex polygonal domain to approximate the curved domain. For either constant coefficients or nonconstant coefficients problem, numerical examples are listed to confirm theoretical analysis respectively.

Two-step MPS-MFS ghost point method for solving partial differential equations

The fictitious boundary surrounding the domain is required in the implementation of the method of fundamental solutions (MFS). A similar approach of taking a portion of the centres of radial basis functions (RBFs) outside the domain in the context of the method of particular solutions (MPS) is proposed to further improve the performance of a two-step MPS-MFS method. Since the shape parameter of RBFs is problem dependent, several known procedures on how to determine a good shape parameter are introduced for solving various types of problems. The proposed method is not only highly accurate but also fairly stable due to the use of the particular solutions and fundamental solutions. To demonstrate the effectiveness of the proposed method, five numerical examples in highly complicate and irregular domains, which include second and fourth order elliptic partial differential equations in 2D and 3D, are presented.

Mixed Finite Element Method for Modified Poisson–Nernst–Planck/Navier–Stokes Equations

In this paper, a complete mixed finite element method is developed for a modified Poisson–Nernst–Planck/Navier–Stokes (PNP/NS) coupling system, where the original Poisson equation in PNP system is replaced by a fourth-order elliptic equation to more precisely account for electrostatic correlations in a simplified form of the Landau–Ginzburg-type continuum model. A stabilized mixed weak form is defined for each equation of the modified PNP/NS model in terms of primary variables and their corresponding vector-valued gradient variables, based on which a stable Stokes-pair mixed finite element is thus able to be utilized to discretize all solutions to the entire modified PNP/NS model in the framework of Stokes-type mixed finite element approximation. Semi- and fully discrete mixed finite element schemes are developed and are analyzed for the presented modified PNP/NS equations, and optimal convergence rates in energy norms are obtained for both schemes. Numerical experiments are carried out to validate all attained theoretical results.

Nodal solutions of fourth-order Kirchhoff equations with critical growth in R^N

We consider a class of fourth-order elliptic equations of Kirchhoff type with critical growth in \(R^N\). By using constrained minimization in the Nehari manifold, weestablish sufficient conditions for the existence of nodal (that is, sign-changing) solutions.
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Solutions to a fourth-order elliptic equation with steep potential

In this paper, we consider the following fourth-order elliptic equation with steep potential and concave–convex nonlinearities Δ2u−(1+a∫R3|∇u|2dx)Δu+Vλ(x)u=f(x)|u|q−2u+g(x)|u|p−2u,x∈R3, existence and multiplicity of solutions to above equation are proved by using variational method. The usual Nehari manifold and Pohozaev type manifold do not work here. Our results can be viewed as an extension of the existing results.

Hessian discretisation method for fourth-order semi-linear elliptic equations: applications to the von Kármán and Navier–Stokes models

This paper deals with the Hessian discretisation method (HDM) for fourth-order semi-linear elliptic equations with a trilinear non-linearity. The HDM provides a generic framework for the convergence analysis of several numerical methods, such as the conforming and nonconforming finite element methods (ncFEMs) and methods based on gradient recovery (GR) operators. The Adini ncFEM and GR method, a specific scheme that is based on cheap, local reconstructions of higher-order derivatives from piecewise linear functions, are analysed for the first time for fourth-order semi-linear elliptic equations with trilinear non-linearity. Four properties, namely, the coercivity, consistency, limit-conformity and compactness, enable the convergence analysis in HDM framework that does not require any regularity of the exact solution. Two important problems in applications, namely, the Navier–Stokes equations in stream function vorticity formulation and the von Karman equations of plate bending, are discussed. Results of numerical experiments are presented for the Morley and Adini ncFEMs, and the GR method.

Schrödinger equations in $\mathbb{R}^4$ involving the biharmonic operator with critical exponential growth

In this paper we prove the existence of at least one nontrivial ground state solution for fourth-order elliptic equations of the form Δ2u−Δu+u=K(x)[f(u)+g(u)],x∈ℝ4,u∈H2(ℝ4),(P) where Δ2:=Δ(Δ) is the biharmonic operator, f is a continuous nonnegative function with polynomial growth at infinity, g is a continuous nonnegative function with exponential growth and K is a positive bounded continuous function that can vanish at infinity. Our results complete the analysis made in F. Sani (Comm. Pure Appl. Anal. 12 (2013), 405–428), where the author studied Schrödinger equations involving the biharmonic operator with coercive potentials. Our approach is based on various techniques such as the mountain-pass theorem, Trundinger–Moser type inequalities and compactness results.

Sign-changing solutions for fourth-order elliptic equations of Kirchhoff type with critical exponent

Sign-changing solutions for fourth-order elliptic equations of Kirchhoff type with critical exponent

A class of fourth-order elliptic equations with concave and convex nonlinearities in $\mathbb{R}^N$

In this article, we study the multiplicity of solutions for a class of fourth-order elliptic equations with concave and convex nonlinearities in $\mathbb{R}^N$. Under the appropriate assumption, we prove that there are at least two solutions for the equation by Nehari manifold and Ekeland variational principle, one of which is the ground state solution.

Strong unique continuation property for a class of fourth order elliptic equations with strongly singular potentials

In this paper we prove the strong unique continuation property for a class of fourth order elliptic equations involving strongly singular potentials. Our argument is to establish some Hardy-Rellich type inequalities with boundary terms and introduce an Almgren’s type frequency function to show some doubling conditions for the solutions to the above-mentioned equations.

On the effective properties of composite elastic plate

We consider the fourth-order elliptic equation describing the Kirchhoff-Love model for pure bending of a thin solid symmetric plate under a transverse load. Following the results of Antonić and Balenović (1999, 2000) [4,5], and Burazin, Jankov and Vrdoljak (2018) [12], we show the local character of the set of all possible composites, also called the G-closure, and prove that the set of composites obtained by periodic homogenization is dense in that set. Moreover, we derive expressions for elastic coefficients of composite plate obtained by mixing two materials in thin layers, also known as laminated materials, and for mixing two materials in the low-contrast regime. Additionally, we derive optimal bounds on the effective energy of a composite material, known as Hashin-Shtrikman bounds. In the setting of two-phase isotropic materials in dimension d=2, explicit optimal Hashin-Shtrikman bounds are calculated.

Existence results of positive solutions for Kirchhoff type biharmonic equationvia bifurcation methods

This paper is concerned with the existence of positive solutions for the fourth order Kirchhoff type problem $$ \left\{\begin{array}{ll} \Delta^{2}u-(a+b\int_\Omega \nabla u ^2dx)\triangle u=\lambda f(u(x)),\ \ \text{in}\ \Omega,\\ u=\triangle u=0,\ \ \text{on}\ \partial\Omega,\\ \end{array} \right. $$ where $\Omega\subset \mathbb{R}^{N}$($N\geq 1$) is a bounded domain with smooth boundary $\partial \Omega$, $a>0, b\geq 0$ are constants, $\lambda\in \mathbb{R}$ is a parameter. For the case $f(u)\equiv u$, we use an argument based on the linear eigenvalue problems of fourth order elliptic equations to show that there exists a unique positive solution for all $\lambda>\Lambda_{1,a}$, here $\Lambda_{1,a}$ is the first eigenvalue of the above problem with $b=0$; For the case $f$ is sublinear, we prove that there exists a positive solution for all $\lambda>0$ and no positive solution for $\lambda

Multiple Solutions for Critical Fourth-Order Elliptic Equations of Kirchhoff type

In this paper, we study the existence and multiplicity of solutions for a class of equations involving a nonlocal term and the biharmonic operator with critical exponent. By new techniques, multiplicity results are established together with variational method. It is worth noting that the method of this paper is obviously different from the existing literature.

Existence and multiplicity of solutions for nonlocal fourth-order elliptic equations with combined nonlinearities

This paper is concerned with the following nonlocal fourth-order elliptic problem: {Δ2u−m(∫Ω|∇u|2dx)Δu=a(x)|u|s−2u+f(x,u),x∈Ω,u=Δu=0,x∈∂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} $$\\begin{aligned} \\textstyle\\begin{cases} \\Delta ^{2}u-m(\\int _{\\varOmega } \\vert \\nabla u \\vert ^{2} \\,dx)\\Delta u=a(x) \\vert u \\vert ^{s-2}u+f(x,u), \\quad x\\in \\varOmega , \\\\ u=\\Delta u=0,\\quad x\\in \\partial \\varOmega , \\end{cases}\\displaystyle \\end{aligned}$$ \\end{document} by using the mountain pass theorem, the least action principle, and the Ekeland variational principle, the existence and multiplicity results are obtained.

Second order splitting of a class of fourth order PDEs with point constraints

We formulate a well-posedness and approximation theory for a class of generalised saddle point problems with a specific form of constraints. In this way we develop an approach to a class of fourth order elliptic partial differential equations with point constraints using the idea of splitting into coupled second order equations. An approach is formulated using a penalty method to impose the constraints. Our main motivation is to treat certain fourth order equations involving the biharmonic operator and point Dirichlet constraints for example arising in the modelling of biomembranes on curved and flat surfaces but the approach may be applied more generally. The theory for well-posedness and approximation is presented in an abstract setting. Several examples are described together with some numerical experiments.

Multiplicity of solutions to the generalized extensible beam equations with critical growth

In this paper by variational methods, we study the multiplicity of solutions to the following fourth-order elliptic equations of Kirchhoff type with critical nonlinearity in RN: Δ2u−M∫RN|∇u|2dxΔu+V(x)u=k(x)|u|q−2u+λ|u|2∗∗−2u,x∈RN, where Δ2u=Δ(Δu) is the biharmonic operator, M:R+→R+ is a continuous function, λ>0 is a parameter, V:RN→R+ is a potential function, and k(x) is a nonnegative continuous real valued function satisfying some conditions. The compactness condition is proved by the Lions’ second Concentration-compactness principle and Concentration-compactness principle at infinity.

Infinitely Many High Energy Solutions for a Fourth-Order Equations of Kirchhoff Type in ℝN

In this paper we study the following fourth-order elliptic equations of Kirchhoff type $$\Delta^2u - (a+b\int_{\mathbb{R}^N} | \triangledown u|^2dx)\Delta u + V(x)u=f(x, u), \;\;x\in\mathbb{R}^N,$$ where Δ2 := Δ(Δ) is the biharmonic operator, a, b > 0 are constants, V ∈ C(ℝN, ℝ) and f ∈ C(ℝN × ℝ, ℝ). Under some appropriate assumptions on V(x) and f(x, u), new results on the existence of infinitely many high energy solutions for the above equation are obtained via Symmetric Mountain Pass Theorem.