We construct new explicit proper biharmonic functions on the 3-dimensional Thurston geometries Sol, Nil, S̃L2(R), H 2 × R and

In this paper we give an enclosure for the solution of the biharmonic problem and also for its gradient and Laplacian in the L2-norm, respectively.

In this paper we consider analytical and numerical solutions to the Dirichlet boundary value problem for the biharmonic partial differential equation, on a disk of finite radius in the plane. The physical interpretation of these solutions is that of the harmonic oscillations of a thin, clamped plate. For the linear, fourth-order, biharmonic partial differential equation in the plane, it is well...

where n is the exterior normal direction of ∂Ω. In other words, we look for a “best” way to extend the boundary value φ with the prescribed normal derivative ψ. Typical examples of Ω and N are the unit ball and the unit sphere, respectively. In this case, ψ : ∂Ω → TφN means φ (x) · ψ (x) = 0 for all |x| = 1. With the given Dirichlet data φ, the most natural extension is perhaps the harmonic map...

A time-dependent electromagnetic field creates electron-hole excitations in a Fermi sea at low temperature. We show that the electron-hole pairs can be generated in a controlled way using harmonic and biharmonic time-dependent voltages applied to a quantum contact, and we obtain the probabilities of the pair creations. For a biharmonic voltage drive, we find that the probability of a pair creat...

Abstract. Using Maz’ya type integral identities with power weights, we obtain new boundary estimates for biharmonic functions on Lipschitz and convex domains in R. For n ≥ 8, combined with a result in [S2], these estimates lead to the solvability of the L Dirichlet problem for the biharmonic equation on Lipschitz domains for a new range of p. In the case of convex domains, the estimates allow u...

Abstract: This paper deals with the study of the numerical solution of biharmonic equations in one dimension. Biharmonic equations appear frequently in many areas of engineering and physics representing some phenomena. The solution of such problems have been tackled by many authors. In this paper, a numerical method based on the Adomian decomposition method is introduced for the approximate sol...

We examine the convergence characteristics of a preconditioned Krylov subspace solver applied to the linear systems arising from low-order mixed finite element approximation of the biharmonic problem. The key feature of our approach is that the preconditioning can be realized using any “black-box” multigrid solver designed for the discrete Dirichlet Laplacian operator. This leads to preconditio...

In this paper, the well-known nonconforming Morley element for biharmonic equations in two spatial dimensions is extended to any higher dimensions in a canonical fashion. The general -dimensional Morley element consists of all quadratic polynomials defined on each -simplex with degrees of freedom given by the integral average of the normal derivative on each -subsimplex and the integral average...