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The paper investigates the approximate properties of biharmonic Poisson integrals for the upper half-plane on the classes of Hölder functions in the uniform metric. The exact values for the upper bounds of the deviations of the functions of the Hölder class H1 from the biharmonic Poisson integrals defined in the upper half-plane were found. [ABSTRACT FROM AUTHOR]

In this paper we study the harmonic map heat flow on the euclidean space R d and we show an unconditional uniqueness result for maps with small initial data in the homogeneous Besov space B ˙ p , ∞ d p (R d) where d < p < ∞. As a consequence we obtain decay rates for solutions of the harmonic map flow of the form ‖ ∇ u (t) ‖ L ∞ (R d) ≤ C t − 1 2 . Additionally, under the assumption of a stronger spatial localization of the initial conditions, we show that the temporal decay happens in a self-similar way. We also explain that similar results hold for the biharmonic map heat flow and the semilinear heat equation with a power-type nonlinearity. [ABSTRACT FROM AUTHOR]

In this paper we study an inverse boundary value problem for the biharmonic operator with first order perturbation. Our geometric setting is that of a bounded simply connected domain in the Euclidean space of dimension three or higher. Assuming that the inaccessible portion of the boundary is flat, and we have knowledge of the Dirichlet-to-Neumann map on the complement, we prove logarithmic type stability estimates for both the first and the zeroth order perturbation of the biharmonic operator. [ABSTRACT FROM AUTHOR]

We prove that the (k+d)-th Neumann eigenvalue of the biharmonic operator on a bounded connected d-dimensional (d\ge 2) Lipschitz domain is not larger than its k-th Dirichlet eigenvalue for all k\in \mathbb {N}. For a special class of domains with symmetries we obtain a stronger inequality. Namely, for this class of domains, we prove that the (k+d+1)-th Neumann eigenvalue of the biharmonic operator does not exceed its k-th Dirichlet eigenvalue for all k\in \mathbb {N}. In particular, in two dimensions, this special class consists of domains having an axis of symmetry. [ABSTRACT FROM AUTHOR]

We consider weak solutions of the nonlinear time-fractional biharmonic diffusion equation ∂ t α u + ∂ t β u + u x x x x = h (t , x) | u | p in (0 , ∞) × (0 , 1) subject to the initial conditions u (0 , x) = u 0 (x) , u t (0 , x) = u 1 (x) and the Navier boundary conditions u (t , 1) = u x x (t , 1) = 0 , where α ∈ (0 , 1) , β ∈ (1 , 2) , ∂ t α (resp. ∂ t β ) is the fractional derivative of order α (resp. β) with respect to the time-variable in the Caputo sense, p > 1 and h is a measurable positive weight function. Using nonlinear capacity estimates specifically adapted to the fourth-order differential operator ∂ 4 ∂ x 4 , the domain, the initial conditions and the boundary condition, a general nonexistence result is established. Next, some special cases of weight functions h are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we consider a coupled system of two biharmonic equations with damping and source terms of variable-exponent nonlinearities, supplemented with initial and mixed boundary conditions. We establish an existence and uniqueness result of a weak solution, under suitable assumptions on the variable exponents. Then, we show that solutions with negative-initial energy blow up in finite time. To illustrate our theoritical findings, we present two numerical examples. [ABSTRACT FROM AUTHOR]

In this manuscript, we show that at least three weak solutions exist for a non-local $ p(x) $ p (x) -biharmonic problem. The variational method and the critical theorem of Bonanno and Marano (Appl. Anal. 89 (2010), 1–10) are used to prove our main result. [ABSTRACT FROM AUTHOR]

In this paper, we determine the variation formula for the first eigenvalue of (p , q) -biharmonic system on a closed Riemannian manifold. Several monotonic quantities are also derived. [ABSTRACT FROM AUTHOR]

We are concerned with semilinear biharmonic equations of the form where B denotes the unit ball in ℝN, N ≥ 1, λ > 0 is a parameter, ∂/∂ν is the outward normal derivative, f : ℝ → (0, +∞) is a continuous function that has super-linear growth at infinity. We use bifurcation theory, combined with an approximation scheme to establish the existence of an unbounded branch of positive radial solutions, which is bounded in positive λ–direction. If in addition, f satisfies certain subcritical condition, we show that the branch must bifurcate from infinity at λ = 0. [ABSTRACT FROM AUTHOR]

In this paper, we study the following biharmonic Choquard-type problem Δ 2 u - β Δ u = λ u + (I μ ∗ F (u)) f (u) , in R 4 , ∫ R 4 | u | 2 d x = c 2 > 0 , u ∈ H 2 (R 4) , where β ≥ 0 , λ ∈ R , I μ = 1 | x | μ with μ ∈ (0 , 4) , F(u) is the primitive function of f(u), and f is a continuous function with exponential critical growth. By using the mountain-pass argument, we prove the existence of radial ground-state solutions for the above problem. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of a ground state solution for a logarithmic weighted biharmonic equation under Dirichlet boundary conditions in the unit ball B$$ B $$ of ℝ4$$ {\mathrm{\mathbb{R}}}&#x0005E;4 $$. The reaction term of the equation is assumed to have exponential growth, in view of Adams' type inequalities. It is proved that there is a ground state solution using min‐max techniques and the Nehari method. The associated energy functional loses compactness at a certain level. An appropriate asymptotic condition allows us to bypass the non‐compactness levels of the functional. [ABSTRACT FROM AUTHOR]

We show that the biharmonic Hilbert complex with mixed boundary conditions on bounded strong Lipschitz domains is closed and compact. The crucial results are compact embeddings that follow by abstract arguments using functional analysis together with particular regular decompositions. Higher Sobolev order results are also proved. [ABSTRACT FROM AUTHOR]

In this paper, we study ultra-weak discontinuous Galerkin methods with generalized numerical fluxes for multi-dimensional high order partial differential equations on both unstructured simplex and Cartesian meshes. The equations we consider as examples are the nonlinear convection-diffusion equation and the biharmonic equation. Optimal error estimates are obtained for both equations under certain conditions, and the key step is to carefully design global projections to eliminate numerical errors on the cell interface terms of ultra-weak schemes on general dimensions. The well-posedness and approximation capability of these global projections are obtained for arbitrary order polynomial space based on a wide class of generalized numerical fluxes on regular meshes. These projections can serve as general analytical tools to be naturally applied to a wide class of high order equations. Numerical experiments are conducted to demonstrate these theoretical results. [ABSTRACT FROM AUTHOR]

This article introduces a unified mixed finite element framework based on a saddle-point formulation that applies to time-dependent fourth order linear and nonlinear problems with clamped, simply supported, and Cahn-Hilliard type boundary conditions. The classical mixed formulations lead to large matrix systems that demand huge storage and computational time making the schemes expensive, especially for the time-dependent problems. The proposed scheme circumvents this by employing biorthogonal basis functions that lead to sparse and positive-definite systems. The article discusses a mixed finite element method for the biharmonic problem and the time-dependent linear and nonlinear versions of the extended Fisher-Kolmogorov equations equipped with the aforementioned boundary conditions. The wellposedness of the scheme is discussed and a priori error estimates are presented for the semi-discrete and fully discrete finite element schemes. The numerical experiments validate the theoretical estimates derived in the paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider the weighted m -biharmonic equation with nonlinear damping and source terms. We proved the global existence of solutions. Later, the decay of the energy is established by using Nakao's inequality. Finally, we proved the blow-up of solutions in finite time. [ABSTRACT FROM AUTHOR]

We prove that the biharmonic NLS equation Δ 2 u + 2 Δ u + (1 + ε) u = | u | p - 2 u i n R d has at least k + 1 geometrically distinct solutions if ε > 0 is small enough and 2 < p < 2 ⋆ k , where 2 ⋆ k is an explicit critical exponent arising from the Fourier restriction theory of O (d - k) × O (k) -symmetric functions. This extends the recent symmetry breaking result of Lenzmann–Weth (Symmetry breaking for ground states of biharmonic NLS via Fourier extension estimates, 2023) and relies on a chain of strict inequalities for the corresponding Rayleigh quotients associated with distinct values of k. We further prove that, as ε → 0 + , the Fourier transform of each ground state concentrates near the unit sphere and becomes rough in the scale of Sobolev spaces. [ABSTRACT FROM AUTHOR]

We consider the inverse random potential scattering problem for the two- and threedimensional biharmonic wave equation in lossy media. The potential is assumed to be a microlocally isotropic Gaussian rough field. The main contributions of the work are twofold. First, the unique continuation principle is proved for the fourth order biharmonic wave equation with rough potentials, and the well-posedness of the direct scattering problem is established in the distribution sense. Second, the correlation strength of the random potential is shown to be uniquely determined by the high frequency limit of the second moment of the backscattering data averaged over the frequency band. Moreover, we demonstrate that the expectation in the data can be removed and the data of a single realization is sufficient for the uniqueness of the inverse problem with probability one when the medium is lossless. [ABSTRACT FROM AUTHOR]

In this paper, we develop a collocation method for solving three-dimensional partial differential equations using Haar wavelet and Kronecker tensor product. The approach is based on a series of Haar wavelet basis functions to approximate sixth-order mixed derivatives. The proposed method is mathematically fast, less error and straightforward for the numerical solution of many types of three-dimensional Poisson, biharmonic and Helmholtz equations. Some numerical examples verify the accuracy and efficiency of the proposed method. Finally, we conclude that numerical results computed by our proposed method are more accurate than numerical results obtained in the existing methods in the literature. We find that the CPU time consumed by the suggested approach is lesser than the CPU time of existing methods. Thus, the process is fast, efficient and has a low numerical error. [ABSTRACT FROM AUTHOR]

An abstract property (H) is the key to a complete a priori error analysis in the (discrete) energy norm for several nonstandard finite element methods in the recent work [Lowest-order equivalent nonstandard finite element methods for biharmonic plates, Carstensen and Nataraj, M2AN, 2022]. This paper investigates the impact of (H) to the a posteriori error analysis and establishes known and novel explicit residual-based a posteriori error estimates. The abstract framework applies to Morley, two versions of discontinuous Galerkin, C0 interior penalty, as well as weakly over-penalized symmetric interior penalty schemes for the biharmonic equation with a general source term in H−2 (Ω). [ABSTRACT FROM AUTHOR]

We develop structure-preserving incomplete LU type factorizations for preconditioning centrosymmetric matrices and use them to numerically solve centrosymmetric and nearly centrosymmetric linear systems arising from spectral methods for partial differential equations. Our algorithm builds in part on direct solution techniques previously developed for this type of linear systems, featuring double-cone factorizations. We illustrate our findings on discretizations of model problems involving the Poisson, diffusion, Helmholtz, and biharmonic equations in one, two, and three dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we address the Robin inverse problem for the biharmonic equation in a 2D$$ 2D $$ simply connected domain, to reconstruct the geometric shape of a non‐accessible part of the boundary from a single measurement of Riquier–Neumann data on the accessible part of that boundary. Our approach extends the nonlinear boundary integral equation, to recover the shape of the boundary. We propose the Newton iterative technique based on the Fréchet derivatives to linearize the system and then establish an injectivity of the linearized system for certain Robin coefficients, as well as the iteration scheme to describe the inverse algorithm for recovering the shape with respect to the unknowns. The mathematical spirit of the proposed method will be presented, and to illustrate its feasibility, some numerical examples will be provided. [ABSTRACT FROM AUTHOR]

This paper deals with an improvement of a class of Adams type inequalities involving potentials V$$ V $$ and weights K$$ K $$, which can decay to zero at infinity as (1+|x|α)−1$$ {\left(1&amp;#x0002B;{\left&amp;#x0007C;x\right&amp;#x0007C;}&amp;#x0005E;{\alpha}\right)}&amp;#x0005E;{-1} $$, α∈(0,4)$$ \alpha \in \left(0,4\right) $$, and (1+|x|β)−1$$ {\left(1&amp;#x0002B;{\left&amp;#x0007C;x\right&amp;#x0007C;}&amp;#x0005E;{\beta}\right)}&amp;#x0005E;{-1} $$, β∈[α,+∞)$$ \beta \in \left[\alpha, &amp;#x0002B;\infty \right) $$, respectively. As an application of this result and by using minimax methods, we establish the existence of solutions for the following class of problems: Δ2u−Δu+V(x)u=K(x)f(x,u)inℝ4,$$ {\Delta}&amp;#x0005E;2u-\Delta u&amp;#x0002B;V(x)u&amp;#x0003D;K(x)f\left(x,u\right)\kern0.30em \mathrm{in}\kern0.30em {\mathrm{\mathbb{R}}}&amp;#x0005E;4, $$where the nonlinear term f(x,u)$$ f\left(x,u\right) $$ can have critical exponential growth. Furthermore, when α∈(0,2)$$ \alpha \in \left(0,2\right) $$ we prove that the solutions belong to the Sobolev space H2ℝ4$$ {H}&amp;#x0005E;2\left({\mathrm{\mathbb{R}}}&amp;#x0005E;4\right) $$ (bound state solutions). [ABSTRACT FROM AUTHOR]

This article studies (optimal) W2m-1,∞-regularity for the polyharmonic equation (-Δ)mu = Q Hn-1 ... Γ, where Γ is a (suitably regular) (n - 1)-dimensional submanifold of ℝn, Hn-1 is the Hausdorff measure, and Q is some suitably regular density. As an application, we derive (optimal) W3, ∞-regularity for solutions of the biharmonic Alt-Caffarelli problem in two dimensions. [ABSTRACT FROM AUTHOR]

We study the following Lane–Emden system: $$\begin{align*} & -\Delta u=|v|^{q-1}v \quad \ \textrm{in}\ \Omega, \qquad -\Delta v=|u|^{p-1}u \quad \ \textrm{in}\ \Omega, \qquad u_{\nu}=v_{\nu}=0 \quad \ \textrm{on}\ \partial \Omega, \end{align*}$$ with |$\Omega $| a bounded regular domain of |${\mathbb{R}}^{N}$|⁠ , |$N \ge 4$|⁠ , and exponents |$p, q$| belonging to the so-called critical hyperbola |$1/(p+1)+1/(q+1)=(N-2)/N$|⁠. We show that, under suitable conditions on |$p, q$|⁠ , least-energy (sign-changing) solutions exist, and they are classical. In the proof we exploit a dual variational formulation, which allows to deal with the strong indefinite character of the problem. We establish a compactness condition which is based on a new Cherrier-type inequality. We then prove such condition by using as test functions the solutions to the system in the whole space and performing delicate asymptotic estimates. If |$N \ge 5$|⁠ , |$p=1$|⁠ , the system above reduces to a biharmonic equation, for which we also prove existence of least-energy solutions. Finally, we prove some partial symmetry and symmetry-breaking results in the case |$\Omega $| is a ball or an annulus. [ABSTRACT FROM AUTHOR]

Let V be a bounded potential. We prove a Hölder stability estimate of determining the potential V from the boundary spectral data of the biharmonic operator Δ 2 + V. The boundary spectral data consists of the Dirichlet eigenvalues λ k and the normal derivatives of the eigenfunctions ∂ ν ϕ k , ∂ ν Δ ϕ k on the boundary of a bounded domain. The analysis depends on the asymptotic behavior of the spectral data. [ABSTRACT FROM AUTHOR]

Boundary-value problems are considered for harmonic and biharmonic equations, as well as the general polyharmonic equation for multiply connected domains on a plane. The problems are reduced to solving linear integral equations at boundary contours, which are assumed to be planar. An algorithm for deriving an approximation of integral equations by a linear system is presented, taking into account the logarithmic singularities of the kernels of integral operators, through which the integral equations are expressed. The algorithm uses the periodicity of functions defined for closed boundary contours. As the number of grid points increases, the approximation error decreases faster than the grid spacing to any fixed power. Applications to solving problems of hydrodynamics, filtration, and other problems of theoretical physics are considered. [ABSTRACT FROM AUTHOR]

We address the Sobolev–Neumann problem for the bi-harmonic equation describing the bending of the Kirchhoff plate with a traction-free edge but fixed at two rows of points. The first row is composed of points placed at the edge, at a small distance ε > 0 between them, and the second one is composed of points placed along a contour at distance O(ε1+α) from the edge. We prove that in the case where α ∈ [0, 1/2), the limit passage as ε → +0 leads to the plate rigidly clamped along the edge while, in the case where α > 1/2, under additional conditions, the limit boundary conditions become of the hinge support type. Based on the asymptotic analysis of the boundary layer in a similar problem, we predict that in the critical case α = 1/2, the boundary hinge-support conditions with friction occur in the limit. We discuss the available generalization of the results and open questions. [ABSTRACT FROM AUTHOR]

This paper is concerned with a backward problem of a stochastic partial differential equation with bi-harmonic operator. The source term is driven by fractional Brownian motion. Based on the Gevrey-type space, the regularity of the mild solution is studied. However, this problem is ill-posed since it is unstable. The instability is discussed in the sense of expectation and variance. Moreover, a regularization method is proposed. The error estimation between the regularization solution and the mild solution is given using an a prior parameter choice rule. [ABSTRACT FROM AUTHOR]

In this paper, we consider a coupled system of hyperbolic and biharmonic-wave equations with variable exponents in the damping and coupling terms. In each equation, the damping term is modulated by a time-dependent coefficient a(t) (or b(t)). First, we state and prove a well-posedness theorem of global weak solutions, by exploiting Galerkin's method and some compactness arguments. Then, using the multiplier method, we establish the decay rates of the solution energy, under suitable assumptions on the time-dependent coefficients and the range of the variable exponents. We end our work with some illustrative examples. [ABSTRACT FROM AUTHOR]

We prove that the solutions of the Schrödinger and biharmonic Schrödinger equations do not have the exact boundary controllability property on the half-line by showing that the associated adjoint models lack observability. We consider the framework of |$L^2$| boundary controls with data spaces |$H^{-1}(\mathbb{R}_+)$| and |$H^{-2}(\mathbb{R}_+)$| for the classical and biharmonic Schrödinger equations, respectively. The lack of controllability on the half-line contrasts with the corresponding dynamics on a finite interval for a similar regularity setting. Our proof is based on an argument that uses the sharp fractional time trace estimates for solutions of the adjoint models. We also make several remarks on the connection of controllability and temporal regularity of spatial traces. [ABSTRACT FROM AUTHOR]

We apply the Murnaghan-Nakayama rule in the representation theory of symmetric groups to develop new techniques for studying biharmonic hypersurfaces in a space form. As applications of the new techniques, we settle the well-known Chen's conjecture on biharmonic hypersurfaces in \Bbb R^6 and BMO conjecture on biharmonic hypersurfaces in \mathbb S^6. [ABSTRACT FROM AUTHOR]

Boundary-value problems are considered for harmonic and biharmonic equations, as well as the general polyharmonic equation for multiply connected domains on a plane. The problems are reduced to solving linear integral equations at boundary contours, which are assumed to be planar. An algorithm for deriving an approximation of integral equations by a linear system is presented, taking into account the logarithmic singularities of the kernels of integral operators, through which the integral equations are expressed. The algorithm uses the periodicity of functions defined for closed boundary contours. As the number of grid points increases, the approximation error decreases faster than the grid spacing to any fixed power. Applications to solving problems of hydrodynamics, filtration, and other problems of theoretical physics are considered. [ABSTRACT FROM AUTHOR]

We prove a Brezis–Kato-type regularity result for weak solutions to the biharmonic nonlinear equation $$ \begin{align*} & \Delta^2 u = g(x,u)\qquad\text{in }\mathbb{R}^N\end{align*}$$ with a Carathéodory function |$g:\mathbb {R}^N\times \mathbb {R}\to \mathbb {R}$|⁠ , |$N\geq 5$|⁠. The regularity results give rise to the existence of ground state solutions provided that |$g$| has a general subcritical growth at infinity. We also conceive a new biharmonic logarithmic Sobolev inequality $$ \begin{align*} & \int_{\mathbb{R}^N}|u|^2\log |u|\, \text{d}x\leq\frac{N}{8}\log \Big(C\int_{\mathbb{R}^N}|\Delta u|^2\, \text{d}x \Big), \quad\text{for } u \in H^2(\mathbb{R}^N), \; \int_{\mathbb{R}^N}u^2\, \text{d}x = 1, \end{align*}$$ for a constant |$0

A method is proposed for building exact solutions to boundary value problems of the theory of elasticity in a rectangle with stiffeners located inside a region (the inhomogeneous problem). The solutions are presented as series in Papkovich–Fadle eigenfunctions with explicitly determined coefficients. The method is based on the Papkovich orthogonality relation and the developed theory of expansions in the Papkovich–Fadle eigenfunctions in homogeneous boundary value problems of the theory of elasticity in a rectangle (the biharmonic problem). The solution sequence is demonstrated by the example of an even symmetric problem for a rectangle in which the sides are free and an external load acts along a stiffener located on the axis of symmetry of the rectangle. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the existence of multiple weak solutions of the following fourth-order nonlinear elliptic problem with a p (x) -biharmonic operator { Δ p (x) 2 u = λ f (x) | u | q (x) − 2 u , x ∈ Ω , ∂ (| Δ u | p (x) − 2 Δ u) ∂ n = g (x) | u | r (x) − 2 u , x ∈ ∂ Ω , where Ω ⊂ R N is a bounded domain, Δ p (x) 2 u = Δ (| Δ u | p (x) − 2 Δ u) is the operator of fourth order called the p (x) -biharmonic operator, p (x) , q (x) , r (x) ∈ C (Ω ¯) , and f ∈ C (Ω ¯) , g ∈ C (∂ Ω) are non-negative weight functions with compact support in Ω. Our analysis mainly relies on variational arguments and some recent theory on the generalized Lebesgue–Sobolev spaces L p (x) (Ω) and W m , p (x) (Ω). [ABSTRACT FROM AUTHOR]

A conforming discontinuous Galerkin finite element method is introduced for solving the biharmonic equation. This method, by its name, uses discontinuous approximations and keeps simple formulation of the conforming finite element method at the same time. The ultra simple formulation of the method will reduce programming complexity in practice. Optimal order error estimates in a discrete H² norm is established for the corresponding finite element solutions. Error estimates in the L² norm are also derived with a sub-optimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence. [ABSTRACT FROM AUTHOR]

We study the existence of multiple solutions for a biharmonic problem Δ 2 u = f x , u + g x , u in Ω , u = ∂ v u = 0 on ∂ Ω , where Ω is a bounded domain with smooth boundary in ℝN, N > 4, f(x, ξ) is odd in ξ, and g(x, ξ) is a perturbation term. Under certain growth conditions on f and g, we show that there are infinitely many weak solutions to the problem. [ABSTRACT FROM AUTHOR]

In this paper, stability analysis and error estimates of a fully discrete local discontinuous Galerkin method for solving the linear bi-harmonic equation are carried out, where the time discretization is treated by a strong-stability-preserving implicit Runge-Kutta scheme. Based on the generalized alternating numerical fluxes, the relationship between the numerical solution and the inner product of the auxiliary solution is established, which plays a key role in obtaining the unconditional stability of the fully discrete local discontinuous Galerkin methods. By carefully introducing reference functions and generalized Gauss-Radau projection, the optimal error estimates are obtained. Numerical experiments are given to demonstrate the stability and accuracy of the fully discrete scheme for the linear bi-harmonic equation. [ABSTRACT FROM AUTHOR]

This article is devoted to the construction of pseudomodes of one-dimensional biharmonic operators with complex-valued potentials via the WKB method. As a by-product, the shape of pseudospectrum near infinity can be described. This is a newly discovered systematic method that goes beyond the standard semiclassical setting which is a direct consequence. This approach can cover a wide class of previously inaccessible potentials, from logarithmic to superexponential ones. [ABSTRACT FROM AUTHOR]

The stream function solution for the inner region Stokes flow, for a locally plane moving fluid interface near the triple point, is derived considering three different boundary conditions: the Navier slip boundary condition (NBC), the super-slip boundary condition and the generalized Navier boundary condition (GNBC). The NBC, incorporating a slip length parameter λ , is a well-known method for regularization in the context of the three-phase dynamic contact line problem. It is demonstrated that the velocity field solution under this boundary condition maintains a C0 continuity at the contact line, resulting in a logarithmic divergence of the pressure at the contact line. By contrast, the super-slip boundary condition establishes a proportional relationship between the wall velocity and the normal derivative of the shear stress, leading to a C1 velocity field. Furthermore, the GNBC, which introduces an uncompensated Young stress to drive the contact line, yields a C2 velocity field. The dominant terms are explicitly derived, and the analytical approach presented here can be extended to other bi-harmonic problems as well. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following fourth order elliptic equation Δ 2 u - Δ u + u = κ (x) | u | q - 2 u + | u | p - 2 u , where Δ 2 : = Δ (Δ) is the biharmonic operator, u ∈ H 2 (R N) , N > 4 , 1 < q < 2 < p < 2 ∗ = 2 N N - 4 , and the weight function κ may change sign in R N . By using the Nehari manifold method, we obtain two distinct solutions to this problem under the predefined conditions of κ . [ABSTRACT FROM AUTHOR]

This paper is devoted to a class of general nonlocal fourth-order elliptic equation Δ 2 u − (a + b ∫ R N | ∇ u | 2 d x) Δ u + V (x) u = f (x , u) , x ∈ R N , where Δ 2 = Δ (Δ) is the biharmonic operator, a > 0 , b ≥ 0 are constants and the potential V is allowed to be sign-changing. Under certain assumptions on f, infinitely many high energy solutions are established. Our results enrich that obtained in the literature. [ABSTRACT FROM AUTHOR]

The notion of k-harmonic curves is associated with the kth-order variational problem defined by the k-energy functional. The present paper gives a geometric formulation of this higher-order variational problem on a Riemannian manifold M and describes a generalized Legendre transformation defined from the kth-order tangent bundle T k M to the cotangent bundle T * T k − 1 M . The intrinsic version of the Euler–Lagrange equation and the corresponding Hamiltonian equation obtained via the Legendre transformation are achieved. Geodesic and cubic polynomial interpolation is covered by this study, being explored here as harmonic and biharmonic curves. The relationship of the variational problem with the optimal control problem is also presented for the case of biharmonic curves. [ABSTRACT FROM AUTHOR]

In this paper, we study the exact boundary controllability of the linear fourth-order Schrödinger equation, with variable physical parameters and clamped boundary conditions on a bounded interval. The control acts on the first spatial derivative at the right endpoint. We prove that this control system is exactly controllable at any time T > 0 . The proofs are based on a detailed spectral analysis and the use of nonharmonic Fourier series. [ABSTRACT FROM AUTHOR]

We investigate the existence of non-trivial weak solutions for the following p(x)-Kirchhoff bi-nonlocal elliptic problem driven by both p(x)-Laplacian and p(x)-Biharmonic operators { M (σ) ( Δ p (x) 2 u - Δ p (x) u) = λ ϑ (x) | u | q (x) - 2 u ( ∫ Ω ϑ (x) q (x) | u | q (x) d x ) r in Ω , u ∈ W 2 , p (.) (Ω) ∩ W 0 1 , p (.) (Ω) , under some suitable conditions on the continuous functions p, q, the non-negative function ϑ and M(σ), where σ : = ∫ Ω | Δ u | p (x) p (x) + | ∇ u | p (x) p (x) d x. Our main results is obtained by employing variational techniques and the well-known symmetric mountain pass lemma. [ABSTRACT FROM AUTHOR]

In this work, we have designed conforming and nonconforming virtual element methods (VEM) to approximate non-stationary nonlocal biharmonic equation on general shaped domain. By employing Faedo–Galerkin technique, we have proved the existence and uniqueness of the continuous weak formulation. Upon applying Brouwer's fixed point theorem, the well-posedness of the fully discrete scheme is derived. Further, following [J. Huang and Y. Yu, A medius error analysis for nonconforming virtual element methods for Poisson and biharmonic equations, J. Comput. Appl. Math. 386 (2021) 113229], we have introduced Enrichment operator and derived a priori error estimates for fully discrete schemes on polygonal domains, not necessarily convex. The proposed error estimates are justified with some benchmark examples. [ABSTRACT FROM AUTHOR]