We study biharmonic hypersurfaces in a generic Riemannian manifold. We first derive an invariant equation for such hypersurfaces generalizing the biharmonic hypersurface equation in space forms studied in [16], [8], [6], [7]. We then apply the equation to show that the generalized Chen’s conjecture is true for totally umbilical biharmonic hypersurfaces in an Einstein space, and construct a (2-p...

In this paper, through making careful analysis of Gauss and Codazzi equations, we prove that four dimensional biharmonic hypersurfaces in nonzero space form have constant mean curvature. Our result gives the positive answer to conjecture proposed by Balmus-Montaldo-Oniciuc 2008 for hypersurfaces.

A multigrid preconditioning scheme for solving the Ciarlet-Raviart mixed method equations for the biharmonic Dirichlet problem is presented. In particular, a Schur complement formulation for these equations which yields non-inherited quadratic forms is considered. The preconditioning scheme is compared with a standard W-cycle multigrid iteration. It is proved that a Variable V-cycle preconditio...

In general, higher order elliptic equations and boundary value problems like the biharmonic equation or the linear clamped plate boundary value problem do not enjoy neither a maximum principle nor a comparison principle or – equivalently – a positivity preserving property. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem from being written ...

For inpainting with linear partial differential equations (PDEs) such as homogeneous or biharmonic diffusion, sophisticated data optimisation strategies have been found recently. These allow high-quality reconstructions from sparse known data. While they have been explicitly developed with compression in mind, they have not entered actual codecs so far: Storing these optimised data efficiently ...

We develop a simple and efficient FFT-based fast direct solver for the biharmonic equation on a disk. The biharmonic equation is split into a coupled system of harmonic problems. We first use the truncated Fourier series expansion to derive a set of coupled singular ODEs, then we solve those singular equations by second-order finite difference discretizations. Using a radial grid with shifting ...

In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem neither a maximum principle nor a comparison principle or – equivalently – a positivity preserving property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem from being rea...

In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem neither a maximum principle nor a comparison principle or – equivalently – a positivity preserving property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem from being rea...

We construct infinitely many real-valued, time-periodic breather solutions of power-type nonlinear wave equations. These are obtained from critical points a dual functional and they weakly localized in space. Our abstract framework allows to find similar existence results for the Klein-Gordon equation or biharmonic

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...