We study the eigenvalues of the biharmonic operators and the buckling eigenvalue on complete, open Riemannian manifolds. We show that the first eigenvalue of the biharmonic operator on a complete, parabolic Riemannian manifold is zero. We give a generalization of the buckling eigenvalue and give applications to studying the stability of minimal Lagrangian submanifolds in Kähler manifolds. MSC 1...

In this article we deal with the existence questions to the nonlinear biharmonic systems. Using theory of monotone operators, we show the existence of a unique weak solution to the weighted biharmonic systems. We also show the existence of a positive solution to weighted biharmonic systems in the unit ball in Rn , using Leray Schauder fixed point theorem. In this study we allow sign-changing we...

The paper studies boundary integral operators of the bi{Laplacian on piecewise smooth curves with corners and describes their mapping properties in the trace spaces of variational solutions of the biharmonic equation. We formulate a direct integral equation method for solving mixed boundary value problems for the biharmonic equation on a nonsmooth plane domain, analyse the solvability of the co...

In this paper, we formulate a novel trivariate biharmonic B-spline defined over bounded volumetric domain. The properties of bi-Laplacian have been well investigated, but the straightforward generalization from bivariate case to trivariate one gives rise to unsatisfactory discretization, due to the dramatically uneven distribution of neighbouring knots in 3D. To ameliorate, our original idea is...

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...

Abstract This article offers a study of the Calderón type inverse problem determining up to second order coefficients higher elliptic operators. Here we show that it is possible determine an anisotropic perturbation given by symmetric matrix, along with first vector field and zero-th potential function inside bounded domain, measuring Dirichlet Neumann map perturbed biharmonic operator on bound...

The existence of generalized Lagrange multipliers is proved for a class evolution problems linear differential operators various types subject to constraints on the derivatives. Those and respective solutions are stable vanishing coercive parameter naturally associated with variational inequalities time-dependent convex sets gradient type. These results applied sandpile problem, superconductivi...

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...