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UDC 515.12 We investigate biharmonic Ricci soliton hypersurfaces $(M^n, g,\xi, \lambda)$ whose potential field $\xi$ satisfies certain conditions. We obtain a result based on the average scalar curvature of the compact Ricci soliton hypersurface $M^n$ where $\xi$ is a general vector field. Then we prove that there are no proper biharmonic Ricci soliton hypersurfaces in the Euclidean space $E^{n+1}$ provided that the potential field $\xi$ is either a principal vector in grad $H^\perp$ or $\xi=\dfrac{{ \rm{ grad } \,} H}{|{ \rm{ grad } \,} H|}$.

Differential equations with variable exponent arise from the nonlinear elasticity theory and the theory of electrorheological fluids. We study the existence of at least three weak solutions for the nonlocal elliptic problems driven by p ⁢ ( x ) p(x) -biharmonic operator. Our technical approach is based on variational methods. Some applications illustrate the obtained results. We also provide an example in order to illustrate our main abstract results. We extend and improve some recent results.

In this paper, we consider the following critical p-biharmonic equation involving Hardy potential $$\begin{aligned} \begin{aligned} \Delta _p^2u-\Delta _qu-\mu \frac{|u|^{p-2}u}{|x|^{2p}}=|u|^{p_{*}-2}u,\quad {{x\in \mathbb {R}^{N}}}, \end{aligned} \end{aligned}$$ where $$2\leqslant p

In this paper, we establish some Schwarz type lemmas for mappings $$\Phi $$ satisfying the inhomogeneous biharmonic Dirichlet problem $$ \Delta (\Delta (\Phi )) = g$$ in $${\mathbb D}$$ , $$\Phi =f$$ on $${\mathbb T}$$ and $$\partial _n \Phi =h$$ on $${\mathbb T}$$ , where g is a continuous function on $$\overline{{\mathbb D}}$$ , f, h are continuous functions on $${\mathbb T}$$ , where $${\mathbb D}$$ is the unit disc of the complex plane $${\mathbb C}$$ and $${\mathbb T}=\partial {\mathbb D}$$ is the unit circle. To reach our aim, we start by investigating some properties of generalized harmonic functions called $$T_\alpha $$ -harmonic functions. Finally, we prove a Landau-type theorem for this class of functions, when $$\alpha >0$$ .

In this paper, we study the inverse problem of identifying unknown sources for nonhomogeneous Biharmonic Equations in bounded domain. It shows that the problem is ill-posed. We use Landweber iterative regularization method to recover the ill-posed solution and give the convergence estimates under the priori and posteriori regularization parameter selection rules. Finally, several numerical examples are given to show the effectiveness and stability of this method.

We study the properties of generalized solutions in unbounded domains and the asymptotic behavior of solutions of elliptic boundary value problems at infinity. Namely, we study the unique solvability of the mixed biharmonic problem with the Steklov and Steklov-type conditions on the boundary in the exterior of a compact set under the assumption that generalized solutions of this problem has a bounded Dirichlet integral with weight $$|x|^{a}$$ . For solving this biharmonic problem with application we use the variational principle, and depending on the value of the parameter $$a$$ , we obtained uniqueness (non-uniqueness) theorems or present exact formulas for the dimension of the space of solutions.

We study the uniqueness and the asymptotic expansion of the solution of the mixed Dirichlet–Neumann problem for the biharmonic equation in the exterior of a compact set under the assumption that the generalized solution of this problem has a finite Dirichlet integral with power-law weight with exponent $$a\in \mathbb {R}$$ . Depending on the value of the parameter $$a$$ , uniqueness (nonuniqueness) theorems are proved and exact formulas are found for the dimension of the linear solution space of the mixed Dirichlet–Neumann problem.

In this paper, using the variational principle, the existence and multiplicity of solutions for p x , q x -Kirchhoff type problem with Navier boundary conditions are proved. At the same time, the sufficient conditions for the multiplicity of solutions are obtained.

In this paper, we study biharmonic hypersurfaces in a product Lm×R of an Einstein space Lm and a real line R. We prove that a biharmonic hypersurface with constant mean curvature in such a product is either minimal or a vertical cylinder generalizing a result of [36] and [17]. We derived the biharmonic equation for hypersurfaces in Sm×R and Hm×R in terms of the angle function of the hypersurface, and use it to obtain some classifications of biharmonic hypersurfaces in such spaces. These include classifications of biharmonic hypersurfaces which are totally umbilical or semi‐parallel for m≥3, and some classifications of biharmonic surfaces in S2×R and H2×R which are constant angle or belong to certain classes of rotation surfaces.

Path planning has been widely investigated by many researchers and engineers for its extensive applications in the real world. In this paper, a biharmonic radial basis potential function (BRBPF) representation is proposed to construct navigation fields in 2D maps with obstacles, and it therefore can guide and design a path joining given start and goal positions with obstacle avoidance. We construct BRBPF by solving a biharmonic equation associated with distance-related boundary conditions using radial basis functions (RBFs). In this way, invalid gradients calculated by finite difference methods in large size grids can be preventable. Furthermore, paths constructed by BRBPF are smoother than paths constructed by harmonic potential functions and other methods, and plenty of experimental results demonstrate that the proposed method is valid and effective.

We derive sharp quantitative bounds for eigenvalues of biharmonic operators perturbed by complex-valued potentials in dimensions one, two and three.

Green’s functions for the Navier and Riquier–Neumann problems for the biharmonic equation in the unit ball are constructed, and integral representations of the solution of these problems are given.

In this paper we study the existence and the multiplicity of nontrivial weak solutions for a fourth order variable exponent Kirchhoff type problem involving p(x)-biharmonic operator with changing sign weight and with no flux boundary condition. By using variational approach and the theory of variable exponent Sobolev spaces, we determine an interval of parameters for which this problem admits at least two nontrivial weak solutions.

We prove that a necessary and sufficient condition for the existence of a classical solution of the inhomogeneous biharmonic equation in a bounded plane domain is the requirement of stronger continuity of the function on the right-hand side in the equation.

We propose an algorithm for construction of quadrature formulas for linear operators acting on periodic functions. For analytic functions, the order of accuracy of quadrature formulas grows unboundedly with the growth of the number of the grid nodes. Within rather general restrictions on the kernels of linear operators, an exponential estimate of a quadrature formula is proved. To exemplify, we find quadrature formulas for the calculation of integral operators with logarithmic singularities which are used in the boundary element method to obtain overconvergent numerical schemes for solution of boundary value problems for harmonic and biharmonic equations on the plane.

We establish Picone’s identity for p-Laplace operator and biharmonic operator on hyperbolic space. We consider both, the ball model $${\mathbb {B}}^N$$ and the half space model $$\mathbb {H}^N.$$ As an application of Picone’s identity, we show that the principal eigenvalue $$\lambda _1$$ of $$- \Delta _{p, \mathbb {H}}$$ is simple. We also obtain a Hardy type inequality on hyperbolic space.

Korpinar, Talat/0000-0003-4000-0892 In this paper, we define concept of the uniformly quasi circular motion (UQCM) with biharmonicity condition in the Heisenberg space. That is, we aim to define a new class of UQCM in the three-dimensional Heisenberg space. We further improve an alternative method to find uniformly quasi circular potential electric energy of biharmonic velocity magnetic particles in the Heisenberg space. We also give the relationships between physical and geometrical characterizations of uniformly quasi circular potential electric energy. Finally, we illustrate important figures for uniformly quasi circular potential electric energy with respect to its electric field in the radial direction.

In this paper we consider a severely ill posed problem associated with a two dimensional homogeneous biharmonic equation. By perturbing the original problem and using a two parameters regularization method, we get a stable solution which converges to the solution of the considered problem. Under some priori bound assumptions, different errors estimates for the regularized solution are obtained. These last ones depend on the choice of the space of the exact solution. To show the effectiveness of the proposed regularization method some numerical results are given.

In this paper, the authors have established the necessary and sufficient conditions for the submanifolds of Kaehler product manifolds to be biharmonic. Moreover, the magnitude of scalar curvature for the hypersurfaces in a product of two unit spheres has been derived. Also, for the same product, the magnitude of the mean curvature vector for Lagrangian submanifolds has been estimated. Finally, the non-existence condition for totally complex Lagrangian submanifolds in a product of unit sphere and a hyperbolic space has been proved.

In this article, we study the existence of at least three distinct weak solutions for nonlocal elliptic problems involving $ p(x) $-biharmonic operator. The results are obtained by means of variational methods. We also provide an example in order to illustrate our main abstract results. We extend and improve some recent results.

This paper is concerned with the existence of an eigenvalue for a p(x)-biharmonic Kirchhoff problem with Navier boundary condition. Under some suitable conditions, we establish that any λ > 0 is an eigenvalue . The proofs combine variational methods with energy estimates. The main results of this paper improve and generalize the previous one introduced by Kefi and Radulescu (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 29 (2018), 439-463).

We construct biharmonic non-harmonic maps between Riemannian manifolds (M, g) and (N, h) by first making the ansatz that φ:(M, g) → (N, h) be a harmonic map and then deforming the metric on N by h ˜ α = α h + ( 1 - α ) d f ⊗ d f {\tilde h_\alpha } = \alpha h + \left( {1 - \alpha } \right){\rm{d}}f \otimes {\rm{d}}f to render φ biharmonic, where f is a smooth function with gradient of constant norm on (N, h) and α ∈ (0, 1). We construct new examples of biharmonic non-harmonic maps, and we characterize the biharmonicity of some curves on Riemannian manifolds.

A well-known conjecture of Bang Yen-Chen says that the only biharmonic Euclidean submanifolds are minimal ones. In this paper, we consider an extended condition (namely, $L_1$-biharmonicity) on non-degenerate timelike hypersurfaces of the pseudo-Euclidean space $E_1^4$. A Lorentzian hypersurface $x: M_1^3\rightarrow\E_1^4$ is called $L_1$-biharmonic if it satisfies the condition $L_1^2x=0$, where $L_1$ is the linearized operator associated to the first variation of 2-th mean curvature vector field on $M_1^3$. According to the multiplicities of principal curvatures, the $L_1$-extension of Chen's conjecture is affirmed for Lorentzian hypersurfaces with constant ordinary mean curvature in pseudo-Euclidean space $E_1^4$. Additionally, we show that there is no proper $L_1$-biharmonic $L_1$-finite type connected orientable Lorentzian hypersurface in $E_1^4$.

The paper deals with the existence and multiplicity of nontrivial weak solutions for the p(x)-Kirchhofftype problem $$ {\displaystyle \begin{array}{c}-M\left(\underset{\Omega}{\int}\frac{1}{p(x)}{\left|\Delta u\right|}^{p(x)} dx\right){\Delta}_{p(x)}^2u=f\left(x,u\right)\kern0.6em \mathrm{in}\kern0.48em \Omega, \\ {}u=\Delta u=0\kern0.48em \mathrm{on}\kern0.48em \mathrm{\partial \Omega }.\end{array}} $$ By using the variational approach and the Krasnosel’skii genus theory, we prove the existence and multiplicity of solutions for the p(x)-Kirchhoff-type equation.

We study the unique solvability of the mixed Dirichlet-Neumann problem for the biharmonic equation in the exterior of a compact set under the assumption that solutions of this problem has a bounded Dirichlet integral with weight |x|a. Depending on the value of the parameter a, we obtained uniqueness (non-uniqueness) theorems of the problem or present exact formulas for the dimension of the space of solutions of the mixed Dirichlet-Neumann problem

In this paper, we study the existence of multiple solutions for the following biharmonic problem $$\begin{aligned} \Delta ^2 u= & {} f(x,u) + g(x,u)\quad \hbox {in}\quad \Omega ,\\ u= & {} \Delta u =0 \quad \hbox {on}\quad \partial \Omega , \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^N, (N > 4)$$ is a smooth bounded domain and $$f(x,\xi )$$ is odd in $$\xi , g(x,\xi )$$ is a perturbation term. By using the variant of Rabinowitz’s perturbation method, under some growth conditions on f and g, we show that there are infinitely many weak solutions to the problem.

The problem of the mechanics of cracks in an orthotropic plate is considered. A general solution form is proposed as a generalized Papkovich–Neuber representation for the plane problem of the theory of elasticity of an orthotropic body. This representation allows us to write the general solution in displacements through two vector potentials satisfying the generalized harmonic equations. The dependences between the vector potentials of Papkovich–Neuber and the stress function are presented. It is shown that there is a complex-valued solution form through four analytic functions of complex variables associated with coefficients that are the roots of the characteristic equation corresponding to the generalized biharmonic equation. It is proved the statement that the operator of the problem is written only through conjugate analytic functions of complex variables, and the general solution is written through arbitrary linear combinations of four functions of complex variables. A general form of solutions with a given singularities is presented, including representations for both cracks in Mode I and cracks in Mode II. The conditions are analyzed that make it possible to obtain generalized non-singular solutions for cracks of Mode I and II. Finally, we establish the conditions for the regularization of singular solutions through the solutions of the generalized Helmholtz equations that correspond to a particular version of the gradient theory of elasticity.

We study the unique solvability of the mixed biharmonic problem with the Steklov-type and Farwig conditions on the boundary in the exterior of a compact set under the assumption that generalized solutions of this problem has a bounded Dirichlet integral with weight $$|x|^{a}$$ . Depending on the value of the parameter $$a$$ , we obtained uniqueness (non-uniqueness) theorems of this problem or present exact formulas for the dimension of the space of solutions.

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.

We study deformations of a long (narrow after rescaling) Kirchhoff plate with periodic (rapidly oscillating) boundary. We deduce a limiting system of two ordinary differential equations of orders 4 and 2 which describe the deflection and torsion of a two-dimensional plate in the leading order. We also consider point supports (Sobolev conditions) whose configuration influences the result of homogenizing the biharmonic equation by decreasing the size of the limiting system of differential equations or completely eliminating it. The boundary-layer phenomenon near the end faces of the plate is studied for various ways of fastening as well as for angular junctions of two long plates, possibly by point clamps (Sobolev conjugation conditions). We discuss full asymptotic series for solutions of static problems and the spectral problems of plate oscillations.

In this paper, we first establish five versions of Landau-type theorems for five classes of bounded biharmonic mappings $$F(z)=|z|^2G(z)+H(z)$$ on the unit disk $${\mathbb {D}}$$ with $$G(0)=H(0)=J_F(0)-1=0$$ , which improve the related results of earlier authors. In particular, two versions of those Landau-type theorems are sharp. Then we derive five bi-Lipschitz theorems for these classes of bounded and normalized biharmonic mappings.

We give some classifications of biharmonic hypersurfaces with constant scalar curvature. These include biharmonic Einstein hypersurfaces in space forms, compact biharmonic hypersurfaces with constant scalar curvature in a sphere, and some complete biharmonic hypersurfaces of constant scalar curvature in space forms and in a non-positively curved Einstein space. Our results provide additional cases (Theorem 2.3 and Proposition 2.8) that supports the conjecture that a biharmonic submanifold in a sphere has constant mean curvature, and two more cases that support Chen's conjecture on biharmonic hypersurfaces (Corollaries 2.2,2.7)., 11 pages

This article deals with the existence and multiplicity of weak solutions to nonlinear differential equations involving a general $p(x)-$biharmonic operator ( in particular, $p(x)-$biharmonic operator). Our approach is mainly based on variational arguments.

We study the problem of approximation of functions (ψ, β)-differentiable (in the Stepanets sense) whose (ψ, β)-derivative belongs to the class H𝛼 by biharmonic Poisson integrals in the uniform metric.

In this article, a family of H2-nonconforming finite elements on tetrahedral grids is constructed for solving the biharmonic equation in 3D. In the family, the Pl polynomial space is enriched by some high order polynomials for all l ≥ 3 and the corresponding finite element solution converges at the order l − 1 in H2 norm. Moreover, the result is improved for two low order cases by using P6 and P7 polynomials to enrich P4 and P5 polynomial spaces, respectively. The error estimate is proved. The numerical results are provided to confirm the theoretical findings.

In this paper, we consider the following sublinear biharmonic equations $ \begin{equation*} \Delta^2 u + V\left( x \right)u = K(x)|u|^{p-1}u,\ x\in \mathbb{R}^N, \end{equation*} $ where $N\geq5, ~0 \lt p \lt 1$, and $K, V$ both change sign in $\mathbb{R}^N$. We prove that the problem has infinitely many solutions under appropriate assumptions on $K, V$. To our end, we firstly infer the boundedness of $PS$ sequence, and then prove that the $PS$ condition was satisfied. At last, we verify that the corresponding functional satisfies the conditions of the symmetric Mountain Pass Theorem.

The Morley finite element method (FEM) is attractive for semilinear problems with the biharmonic operator as a leading term in the stream function vorticity formulation of two-dimensional Navier–Stokes problem and in the von Kármán equations. This paper establishes a best-approximation a priori error analysis and an a posteriori error analysis of discrete solutions close to an arbitrary regular solution on the continuous level to semilinear problems with a trilinear nonlinearity. The analysis avoids any smallness assumptions on the data, and so has to provide discrete stability by a perturbation analysis before the Newton–Kantorovich theorem can provide the existence of discrete solutions. An abstract framework for the stability analysis in terms of discrete operators from the medius analysis leads to new results on the nonconforming Crouzeix–Raviart FEM for second-order linear nonselfadjoint and indefinite elliptic problems with $L^\infty $ coefficients. The paper identifies six parameters and sufficient conditions for the local a priori and a posteriori error control of conforming and nonconforming discretizations of a class of semilinear elliptic problems first in an abstract framework and then in the two semilinear applications. This leads to new best-approximation error estimates and to a posteriori error estimates in terms of explicit residual-based error control for the conforming and Morley FEM.

We construct the Green functions of the stationary problems of heat conduction and thermoelasticity for half spaces with free or rigidly, smoothly, or flexibly fastened boundaries subjected to the action of heat dipoles in the case where the temperature of the boundaries is kept equal to zero or the conditions of heat insulation are imposed on these boundaries. In this case, harmonic double-layer potentials are used for the solution of the problem of heat conduction, as well as thermoelastic potentials of displacements in the infinite space with two heat dipoles located symmetrically about the boundary. To satisfy the boundary conditions on the boundary, we construct the Love biharmonic functions. We present explicit expressions for temperature, displacements, and stresses, which can be used for the determination of the thermoelastic state of the half space caused by the perturbation of a given heat flux by a thin heatinsulating inclusion parallel to the boundary.

In this paper a new class of well-posed boundary value problems for the biharmonic equation is studied. The considered problems are nonlocal boundary value problems of Bitsadze- -Samarskii type. These problems are solved by reducing them to Dirichlet and Neumann type problems. Theorems on existence and uniqueness of the solution are proved and exact solvability conditions of the considered problems are found. In addition, the integral representations of solutions are obtained.

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.

We are interested in the qualitative properties of positive entire solutions $u \in C^4 (\mathbb{R}^n \backslash \{0\})$ of the equation \begin{equation} \label{0.0} \Delta^2 u=u^{\frac{n+4}{n-4}} \;\;\mbox{in $\mathbb{R}^n \backslash \{0\}$ and 0 is a non-removable singularity of $u(x)$}. \end{equation} It is known from [Theorem 4.2] that any positive entire solution $u$ of \eqref{0.0} is radially symmetric with respect to $x=0$, i.e. $u(x)=u(|x|)$, and equation \eqref{0.0} also admits a special positive entire solution $u_s (x)=\Big(\frac{n^2 (n-4)^2}{16} \Big)^{\frac{n-4}{8}} |x|^{-\frac{n-4}{2}}$. We first show that $u-u_s$ changes signs infinitely many times in $(0, \infty)$ for any positive singular entire solution $u \not \equiv u_s$ in $\mathbb{R}^N \backslash \{0\}$ of \eqref{0.0}. Moreover, equation \eqref{0.0} admits a positive entire singular solution $u(x) \; (=u(|x|)$ such that the scalar curvature of the conformal metric with conformal factor $u^{\frac{4}{n-4}}$ is positive and $v(t):=e^{\frac{n-4}{2} t} u(e^t)$ is $2T$-periodic with suitably large $T$. It is still open that $v(t):=e^{\frac{n-4}{2} t} u(e^t)$ is periodic for any positive entire solution $u(x)$ of \eqref{0.0}., Comment: 21 pages; comments welcome

We study the discrete spectra of boundary value problems for the biharmonic operator describing oscillations of a Kirchhoff plate in the form of a locally perturbed strip with rigidly clamped or simply supported edges. The two methods are applied: variational and asymptotic. The first method shows that for a narrowing plate the discrete spectrum is empty in both cases, whereas for a widening plate at least one eigenvalue appears below the continuous spectrum cutoff for rigidly clamped edges. The presence of the discrete spectrum remains an open question for simply supported edges. The asymptotic method works only for small variations of the boundary. While for a small smooth perturbation the construction of asymptotics is generally the same for both types of boundary conditions, the asymptotic formulas for eigenvalues can differ substantially even in the main correction term for a perturbation with corner points.

The existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both p(·)-Harmonic and p(·)-biharmonic operators Δ p ( x ) 2 u - Δ p ( x ) u = λ w ( x ) | u | q ( x ) - 2 u in Ω , u ∈ W 2 , p ( ⋅ ) ( Ω ) ∩ W 0 - 1 , p ( ⋅ ) ( Ω ) , \eqalign{& \Delta _{p\left( x \right)}^2u - {\Delta _{p\left( x \right)}}u = \lambda w\left( x \right){\left| u \right|^{q\left( x \right) - 2}}u\,\,\,{\rm{in}}\,\,\Omega {\rm{,}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,u \in {W^{2,p\left( \cdot \right)}}\left( \Omega \right) \cap W_0^{ - 1,p\left( \cdot \right)}\left( \Omega \right), \cr} is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces Lp (·)(Ω) and Wm,p (·)(Ω).

The biharmonic equation arises in areas of continuum mechanics, including mechanics of elastic plates and the slow flow of viscous fluids. In this paper, we make an effort to establish the generalized versions of Lions-type theorem under various conditions and then apply them to study the existence of ground state solutions for the biharmonic equation.

Deep learning—in particular, deep neural networks (DNNs)—as a mesh-free and self-adapting method has demonstrated its great potential in the field of scientific computation. In this work, inspired by the Deep Ritz method proposed by Weinan E et al. to solve a class of variational problems that generally stem from partial differential equations, we present a coupled deep neural network (CDNN) to solve the fourth-order biharmonic equation by splitting it into two well-posed Poisson’s problems, and then design a hybrid loss function for this method that can make efficiently the optimization of DNN easier and reduce the computer resources. In addition, a new activation function based on Fourier theory is introduced for our CDNN method. This activation function can reduce significantly the approximation error of the DNN. Finally, some numerical experiments are carried out to demonstrate the feasibility and efficiency of the CDNN method for the biharmonic equation in various cases.

We first prove that, unlike the biharmonic case, there exist triharmonic curves with nonconstant curvature in a suitable Riemannian manifold of arbitrary dimension. We then give the complete classification of triharmonic curves in surfaces with constant Gaussian curvature. Next, restricting to curves in a 3-dimensional Riemannian manifold, we study the family of triharmonic curves with constant curvature, showing that they are Frenet helices. In the last part, we give the full classification of triharmonic Frenet helices in space forms and in Bianchi-Cartan-Vranceanu spaces., Comment: To appear in Mediterranean Journal of Mathematics