In this paper, we consider a biharmonic equation with respect to the Dirichlet problem on domain of locally finite graph. Using variation method, prove that has two distinct solutions under certain conditions.

Abstract. We consider multigrid algorithms for the biharmonic problem discretized by conforming 1 finite elements. Most finite elements for the biharmonic equation are nonnested in the sense that the coarse finite element space is not a subspace of the space of similar elements defined on a refined mesh. To define multigrid methods, certain intergrid transfer operators have to be constructed. W...

∆p := ∆(|∆u|p−2∆u) is the operator of fourth order, so-called the p-biharmonic (or p-bilaplacian) operator. For p = 2, the linear operator ∆2 = ∆2 = ∆ · ∆ is the iterated Laplacian that to a multiplicative positive constant appears often in the equations of Navier-Stokes as being a viscosity coefficient, and its reciprocal operator noted (∆2)−1 is the celebrated Green’s operator (see [8]). Exis...

In a radar imaging problem using broad-band, low-frequency waves, we encounter the problem of solving Poisson's equation over a very large rectangular grid, typically ve thousand times thousand pixels. In addition, no information about boundary values is available. In order to select suitable solutions we solve the Poisson equation under the side condition that some criterion function, usually ...

In this paper, we consider p-biharmonic submanifolds of aspace form. We give the necessary and sufficient conditions for a submanifold to be in space present some new properties stress p-bienergy tensor.

We classify biharmonic submanifolds with certain geometric properties in Euclidean spheres. For codimension 1, we determine the biharmonic hypersurfaces with at most two distinct principal curvatures and the conformally flat biharmonic hypersurfaces. We obtain some rigidity results for pseudo-umbilical biharmonic submanifolds of codimension 2 and for biharmonic surfaces with parallel mean curva...

Motion of a biharmonic system under action of small periodic force and small damped force is studied. The biharmonic oscillator is a physical system acting under a biharmonic force like a sin θ b sin 2θ. The article contains biharmonic oscillator analysis, phase space research, and analytic solutions for separatrixes. The biharmonic oscillator performs chaotic motion near separatrixes under sma...

We study entire solutions of the biharmonic heat equation on complete Riemannian manifolds without boundary. provide exponential decay estimates for kernel under assumptions lower bound Ricci curvature and noncollapsing unit balls. And we prove a uniqueness criteria Cauchy problem. As corollaries, conservation law uniform $$L^\infty $$ estimate starting with bounded initial data.