Biharmonic Operator Research Articles

Uniform resolvent estimates and absence of eigenvalues of biharmonic operators with complex potentials

We quantify the subcriticality of the bilaplacian in dimensions greater than four by providing explicit repulsivity/smallness conditions on complex additive perturbations under which the spectrum remains stable. Our assumptions cover critical Rellich-type potentials too. As a byproduct we obtain uniform resolvent estimates in weighted spaces. Some of the results are new also in the self-adjoint setting.

High‐frequency stability estimates for the linearized inverse boundary value problem for the biharmonic operator with attenuation on some bounded domains

In this article, high‐frequency stability estimates are explored for the determination of the zeroth‐order perturbation of the biharmonic operator with constant attenuation from the linearized partial Dirichlet‐to‐Neumann map when part of the boundary is inaccessible and flat. The results obtained suggest improvement of the stability with an appropriate choice of frequency.

Linearized inverse problem for biharmonic operators at high frequencies

In this paper, we study the phenomenon of increasing stability in the inverse boundary value problems for the biharmonic equation. The inverse problem is to determine the potential from DtN map. It is a kind of nonlinear inverse problem. By considering a linearized form, we obtain an increasing Lipschitz‐like stability when is large. Furthermore, we extend the discussion to the linearized inverse biharmonic potential problem with attenuation, where an exponential dependence of the attenuation constant is traced in the stability estimate.

Self-adaptive alternating direction method of multiplier for a fourth order variational inequality

We propose an alternating direction method of multiplier for approximation solution of the unilateral obstacle problem with the biharmonic operator. We introduce an auxiliary unknown and augmented Lagrangian functional to deal with the inequality constrained, and we deduce a constrained minimization problem that is equivalent to a saddle-point problem. Then the alternating direction method of multiplier is applied to the corresponding problem. By using iterative functions, a self-adaptive rule is used to adjust the penalty parameter automatically. We show the convergence of the method and give the penalty parameter approximation in detail. Finally, the numerical results are given to illustrate the efficiency of the proposed method.

Sign-changing Solutions for Fourth Order Elliptic Equation with Concave-convex Nonlinearities

In this paper, we study the following fourth order elliptic equation: Δ²u - Δu + V(x)u = κ(x)|u|q-2u + |u|p-2u in RN, where Δ² := Δ(Δ) is the biharmonic operator, 4 > N, 1 < q < 2 < p < 2* := 2N / (N - 4). Assuming that V(x) satisfies a class of coercive conditions and the nonnegative weighted function κ(x) belongs to Lp / (p-q)(RN), we obtain the existence of one sign-changing solution with the help of constraint variational method and quantitative deformation lemma. The novelty of this paper is that when the nonlinearity is the combination of concave and convex functions, we are able to obtain the existence of sign-changing solutions. Some recent results are improved and generalized significantly.

A note on the paper "An inverse problem on determining second order symmetric tensor for perturbed biharmonic operator"

A note on the paper "An inverse problem on determining second order symmetric tensor for perturbed biharmonic operator"

Ground state solutions for weighted biharmonic Kirchhoff problem via the Nehari manifold approach

In this article, we delve into the investigation of the following non-local problem within the unit ball B 1 of R 4 : g ( ∫ B 1 w β ( x ) | Δu | 2 ) Δ w β 2 u = | u | q − 2 u + f ( x , u ) in B 1 , u = ∂ n u = 0 on ∂ B 1 , where Δ w β 2 . = Δ ( w β ( x ) Δ . ) is the weighted Biharmonic operator, where w β ( x ) represents a logarithmically singular weight, and the non-linearity involves a combination of a reaction source f ( x , u ) , critical in the context of Adams' type exponential inequality and a polynomial function. The Kirchhoff function g is characterized by being positive and continuous. Employing the Nehari manifold method, the quantitative deformation lemma and degree theory results, we establish the existence of a ground state solution.

Improved inequalities between Dirichlet and Neumann eigenvalues of the biharmonic operator

We prove that the ( k + d ) (k+d) -th Neumann eigenvalue of the biharmonic operator on a bounded connected d d -dimensional ( d ≥ 2 ) (d\ge 2) Lipschitz domain is not larger than its k k -th Dirichlet eigenvalue for all k ∈ N 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 ) (k+d+1) -th Neumann eigenvalue of the biharmonic operator does not exceed its k k -th Dirichlet eigenvalue for all k ∈ N k\in \mathbb {N} . In particular, in two dimensions, this special class consists of domains having an axis of symmetry.

Stability estimates for an inverse boundary value problem for biharmonic operators with first order perturbation from partial data

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.

Numerical solution of the boundary value problems for the biharmonic equations via quasiseparable representations

AbstractThe paper incorporates new methods of numerical linear algebra for the approximation of the biharmonic equation with potential, namely, numerical solution of the Dirichlet problem for $$ \left( \frac{d}{dx}\right) ^4u(x)+c(x)u(x)=\phi (x),\quad 0<x<1. $$ d dx 4 u ( x ) + c ( x ) u ( x ) = ϕ ( x ) , 0 < x < 1 . High-order discrete finite difference operators are presented, constructed on the basis of discrete Hermitian derivatives, and the associated Discrete Biharmonic Operator (DBO). It is shown that the matrices associated with the discrete operator belong to a class of quasiseparable matrices of low rank matrices. The application of quasiseparable representation of rank structured matrices yields fast and stable algorithm for variable potentials c(x). Numerical examples corroborate the claim of high order accuracy of the algorithm, with optimal complexity O(N).

Multiple Solutions for Problems Involving p(x)-Laplacian and p(x)-Biharmonic Operators

Multiple Solutions for Problems Involving p(x)-Laplacian and p(x)-Biharmonic Operators

Fourth-order modon in a rotating self-gravitating fluid

We present a two-dimensional nonlinear equation to govern the dynamics of disturbances in a rotating self-gravitating fluid. The nonlinear term of the equation has the form of a Poisson bracket (Jacobian), and the linear part contains, along with the Laplacian, a biharmonic operator. A solution was found in the form of a dipole vortex (modon). The solution and all its derivatives up to the fourth order are continuous on the separatrix.

Nonlinear degenerate Navier problem involving the weighted biharmonic operator with measure data in weighted Sobolev spaces

Nonlinear degenerate Navier problem involving the weighted biharmonic operator with measure data in weighted Sobolev spaces

On solvability of some inverse problems for a nonlocal fourth-order parabolic equation with multiple involution

<abstract><p>In this paper, the solvability of some inverse problems for a nonlocal analogue of a fourth-order parabolic equation was studied. For this purpose, a nonlocal analogue of the biharmonic operator was introduced. When defining this operator, transformations of the involution type were used. In a parallelepiped, the eigenfunctions and eigenvalues of the Dirichlet type problem for a nonlocal biharmonic operator were studied. The eigenfunctions and eigenvalues for this problem were constructed explicitly and the completeness of the system of eigenfunctions was proved. Two types of inverse problems on finding a solution to the equation and its righthand side were studied. In the two problems, both of the righthand terms depending on the spatial variable and the temporal variable were obtained by using the Fourier variable separation method or reducing it to an integral equation. The theorems for the existence and uniqueness of the solution were proved.</p></abstract>

Blow-up of solutions for a time fractional biharmonic equation with exponentional nonlinear memory

<p>In the paper, we focus on the local existence and blow-up of solutions for a time fractional nonlinear equation with biharmonic operator and exponentional nonlinear memory in an Orlicz space. We first establish a $ L^p-L^q $ estimate for solution operators of a time fractional nonlinear biharmonic equation, and obtain bilinear estimates for mild solutions. Then, based on the contraction mapping principle, we establish the local existence of mild solutions. Moreover, by using the test function method, we obtain the blow-up result of solutions.</p>

Triple solutions for a Leray-Lions $ p(x) $-biharmonic operator involving Hardy potential and indefinite weight

<p>Existence and multiplicity of three weak solutions for a Leray-Lions $ p(x) $-biharmonic problem involving Hardy potential and indefinite weight were proved. Our main tools combined variational methods and some critical theorems.</p>

Existence of two non-zero weak solutions for a $p(\cdot)$-biharmonic problem with Navier boundary conditions

In this paper, the existence of non-trivial weak solutions for some problems with Navier boundary conditions driven by the p(\cdot) -biharmonic operator is investigated. The proofs combine variational methods with topological arguments.

Infinitely many solutions for problems involving -Laplacian and -biharmonic operators

In this paper, we study by using variational methods, mountain pass theorem, -mountain pass theorem combined with the theory of the generalized Lebesgue Sobolev spaces, the existence and multiplicity of solutions for problems involving -Laplacian and -biharmonic operators with Hardy-type term.

Pseudomodes for Biharmonic Operators with Complex Potentials

Pseudomodes for Biharmonic Operators with Complex Potentials

Stability for an inverse spectral problem of the biharmonic Schrödinger operator

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 Δ2+V. The boundary spectral data consists of the Dirichlet eigenvalues λk and the normal derivatives of the eigenfunctions ∂νϕk,∂νΔϕk on the boundary of a bounded domain. The analysis depends on the asymptotic behavior of the spectral data.