We clarify the validity of a method that decouples a boundary value problem of biharmonic equation to two Poisson equations on polygonal domains. The method provides a way of computing deflections of simply supported polygonal plates by using Poisson solvers. We show that such decoupling is not valid if the polygonal domain is not convex. It may fail even when the right hand side function is in...

A nonlinear boundary value problem involving the p-biharmonic operator is investigated, where p > 1. It describes various problems in the theory of elasticity, e.g. the shape of an elastic beam where the bending moment depends on the curvature as a power function with exponent p− 1. We prove existence of solutions satisfying a quite general boundary condition that incorporates many particular b...

This paper reports a new spectral collocation method for numerically solving two-dimensional biharmonic boundary-value problems. The construction of the Chebyshev approximations is based on integration rather than conventional differentiation. This use of integration allows: (i) the imposition of the governing equation at the whole set of grid points including the boundary points and (ii) the s...

Abstract. We modify the usual biharmonic model, still frequently used in the engineering community to model linearly elastic plates. In its traditional form, the biharmonic model diverges in general, since it does not incorporate shear effects. Our changes make it convergent for all loads, under a usual assumption on how the loads depend on the thickness. The idea is to add “higher order” terms...

Let u be a super-biharmonic function, that is, Δ2u≥ 0, on the unit disc D in the complex plane, satisfying certain conditions. Then it has been shown that u has a representation analogous to the Poisson-Jensen representation for subharmonic functions on D. In the same vein, it is shown here that a function u on any Green domain Ω in a Riemannian manifold satisfying the conditions (−Δ)iu ≥ 0 for...

Abstract. We propose a regularized optimization problem for computing numerical differentiation for the second order derivative for functions with two variables from the noisy values of the function at scattered points, and give the proof of the existence and uniqueness of the solution of this problem. The reconstruction scheme is also given during the proof, which is based on biharmonic Green ...

Using three critical points theorems, we prove the existence of at least three solutions for a quasilinear biharmonic equation.