On conformal biharmonic immersions

This paper studies conformal biharmonic immersions. We first study the transformations of Jacobi operator and the bitension field under conformal change of metrics. We then obtain an invariant equation for a conformal biharmonic immersion of a surface into Euclidean 3-space. As applications, we construct a two-parameter family of non-minimal conformal biharmonic immersions of cylinder into and some examples of conformal biharmonic immersions of four-dimensional Euclidean space into sphere and hyperbolic space, thus providing many simple examples of proper biharmonic maps with rich geometric meanings. These suggest that there are abundant proper biharmonic maps in the family of conformal immersions. We also explore the relationship between biharmonicity and holomorphicity of conformal immersions of surfaces.      

Remarks on biharmonic hypersurfaces in space forms

We consider closed biharmonic hypersurfaces in a Euclidean sphere and prove a rigidity result under a suitable condition on the scalar curvature. Moreover, we establish an integral formula involving the position vector for biharmonic hypersurfaces in space forms. As an application of this formula, we reobtain a result concerning the closed biharmonic hypersurfaces in Euclidean spheres that lie in a closed hemisphere.     

Classification results for biharmonic submanifolds in spheres

We study biharmonic submanifolds of the Euclidean sphere that satisfy certain geometric properties. We classify: (i) the biharmonic hypersurfaces with at most two distinct principal curvatures; (ii) the conformally flat biharmonic hypersurfaces. We obtain some rigidity results for pseudoumbilical biharmonic submanifolds of codimension 2 and for biharmonic surfaces with parallel mean curvature vector field. We also study the type, in the sense of B-Y. Chen, of compact proper biharmonic submanifolds with constant mean curvature in spheres. Dedicated to Professor Vasile Oproiu on his 65th birthday The first author was supported by a INdAM doctoral fellowship, Italy. The second author was supported by PRIN 2005, Italy. The third author was supported by Grant CEEX ET 5871/2006, Romania     

A note on biharmonic submanifolds of product spaces

We investigate biharmonic submanifolds of the product of two space forms. We prove a necessary and sufficient condition for biharmonic submanifolds in these product spaces. Then, we obtain mean curvature estimates for proper-biharmonic submanifold of a product of two unit spheres. We also prove a non-existence result in the case of the product of a sphere and a hyperbolic space.     

Universal bounds for eigenvalues of the biharmonic operator on Riemannian manifolds

In this paper we consider eigenvalues of the Dirichlet biharmonic operator on compact Riemannian manifolds with boundary (possibly empty) and prove a general inequality for them. By using this inequality, we study eigenvalues of the Dirichlet biharmonic operator on compact domains in a Euclidean space or a minimal submanifold of it and a unit sphere. We obtain universal bounds on the (k+1)th eigenvalue on such objects in terms of the first k eigenvalues independent of the domains. The estimate for the (k+1)th eigenvalue of bounded domains in a Euclidean space improves an important inequality obtained recently by Cheng and Yang.     

Boundedness of the gradient of a solution and Wiener test of order one for the biharmonic equation

The behavior of solutions to the biharmonic equation is well-understood in smooth domains. In the past two decades substantial progress has also been made for the polyhedral domains and domains with Lipschitz boundaries. However, very little is known about higher order elliptic equations in the general setting. In this paper we introduce new integral identities that allow to investigate the solutions to the biharmonic equation in an arbitrary domain. We establish: (1) boundedness of the gradient of a solution in any three-dimensional domain; (2) pointwise estimates on the derivatives of the biharmonic Green function; (3) Wiener-type necessary and sufficient conditions for continuity of the gradient of a solution. Mathematics Subject Classification (2000)  35J40, 35J30, 35B65     

On a nonlocal generalization of the biharmonic Dirichlet problem

We consider a mixed problem with the Dirichlet boundary conditions and integral conditions for the biharmonic equation. We prove the existence and uniqueness of a generalized solution in the weighted Sobolev space W ₂². We show that the problem can be viewed as a generalization of the Dirichlet problem.     

The FFTRR-based fast direct algorithms for complex inhomogeneous biharmonic problems with applications to incompressible flows

We develop analysis-based fast and accurate direct algorithms for several biharmonic problems in a unit disk derived directly from the Green’s functions of these problems and compare the numerical results with the “decomposition” algorithms (see Ghosh and Daripa, IMA J. Numer. Anal. 36(2), 824–850 [17]) in which the biharmonic problems are first decomposed into lower order problems, most often either into two Poisson problems or into two Poisson problems and a homogeneous biharmonic problem. One of the steps in the “decomposition algorithm” as discussed in Ghosh and Daripa (IMA J. Numer. Anal. 36(2), 824–850 [17]) for solving certain biharmonic problems uses the “direct algorithm” without which the problem can not be solved. Using classical Green’s function approach for these biharmonic problems, solutions of these problems are represented in terms of singular integrals in the complex z?plane (the physical plane) involving explicitly the boundary conditions. Analysis of these singular integrals using FFT and recursive relations (RR) in Fourier space leads to the development of these fast algorithms which are called FFTRR based algorithms. These algorithms do not need to do anything special to overcome coordinate singularity at the origin as often the case when solving these problems using finite difference methods in polar coordinates. These algorithms have some other desirable properties such as the ease of implementation and parallel in nature by construction. Moreover, these algorithms have O(logN) complexity per grid point where N ² is the total number of grid points and have very low constant behind this order estimate of the complexity. Performance of these algorithms is shown on several test problems. These algorithms are applied to solving viscous flow problems at low and moderate Reynolds numbers and numerical results are presented.     

Proper biharmonic submanifolds in a sphere

In this paper, we obtain a constraint of the mean curvature for proper biharmonic submanifolds in a sphere. We give some characterizations of some proper biharmonic submanifolds with parallel mean curvature vector in a sphere. We also construct some new examples of proper biharmonic submanifolds in a sphere.     

Affine biharmonic submanifolds in 3-dimensional pseudo-Hermitian geometry

The notion of biharmonic map between Riemannian manifolds is generalized to maps from Riemannian manifolds into affine manifolds. Hopf cylinders in 3-dimensional Sasakian space forms which are biharmonic with respect to Tanaka-Webster connection are classified. Dedicated to professor John C. Wood on his 60th birthday.     

Weighted cubic and biharmonic splines

In this paper we discuss the design of algorithms for interpolating discrete data by using weighted cubic and biharmonic splines in such a way that the monotonicity and convexity of the data are preserved. We formulate the problem as a differential multipoint boundary value problem and consider its finite-difference approximation. Two algorithms for automatic selection of shape control parameters (weights) are presented. For weighted biharmonic splines the resulting system of linear equations can be efficiently solved by combining Gaussian elimination with successive over-relaxation method or finite-difference schemes in fractional steps. We consider basic computational aspects and illustrate main features of this original approach.     

A note on the reflection principle for the biharmonic equation and the Stokes system

In this note, we discuss the reflection principle of the Stokes system in a half space for the threedimensional case, and of the biharmonic equation. Admitting different boundary conditions, we use the reflection principle to prove uniqueness of solutions of the Stokes system or the biharmonic equation in weightedLq-spaces     

Conformal and semi-conformal biharmonic maps

We show that a conformal mapping between Riemannian manifolds of the same dimension n ≥ 3 is biharmonic if and only if the gradient of its dilation satisfies a certain second-order elliptic partial differential equation. On an Einstein manifold solutions can be generated from isoparametric functions. We characterise those semi-conformal submersions that are biharmonic in terms of their dilation and the fibre mean curvature vector field.      

Radial entire solutions for supercritical biharmonic equations

We prove existence and uniqueness (up to rescaling) of positive radial entire solutions of supercritical semilinear biharmonic equations. The proof is performed with a shooting method which uses the value of the second derivative at the origin as a parameter. This method also enables us to find finite time blow up solutions. Finally, we study the convergence at infinity of smooth solutions towards the explicitly known singular solution. It turns out that the convergence is different in space dimensions n ≤ 12 and n ≥ 13. Financial support by the Vigoni programme of CRUI (Rome) and DAAD (Bonn) is gratefully acknowledged.     

Remarks on biharmonic maps into spheres

We prove an apriori estimate in Morrey spaces for both intrinsic and extrinsic biharmonic maps into spheres. As applications, we prove an energy quantization theorem for biharmonic maps from 4-manifolds into spheres and a partial regularity for stationary intrinsic biharmonic maps into spheres.Received: 11 March 2003, Accepted: 18 November 2003, Published online: 25 February 2004     

Polygonal interface problems for the biharmonic operator

We study some boundary value problems on two-dimensional polygonal topological networks, where on each face, the considered operator is the biharmonic operator. The transmission conditions we impose along the edges are inspired by the models introduced by H. Le Dret [13] and Destuynder and Nevers [9]. The boundary conditions on the external edges are the classical ones. This class of problem contains the boundary value problems for the biharmonic equation in a plane polygon (see [3, 11, 12, 18]). Conforming to the classical results cited above, we prove that the weak solution of our problem admits a decomposition into a regular part and a singular part, the latter being a linear combination of singular functions depending on the domain and the considered boundary value problem. Finally, we give the exact formula for the coefficients of these singularities.     

Regions of positivity for polyharmonic Green functions in arbitrary domains

The Green function for the biharmonic operator on bounded domains with zero Dirichlet boundary conditions is in general not of fixed sign. However, by extending an idea of Z. Nehari, we are able to identify regions of positivity for Green functions of polyharmonic operators. In particular, the biharmonic Green function is considered in all space dimensions. As a consequence we see that the negative part of any such Green function is somehow small compared with the singular positive part.

     

Representation for the Green’s function of the Dirichlet problem for polyharmonic equations in a ball

We explicitly construct the Green’s function for the Dirichlet problem for polyharmonic equations in a ball in a space of arbitrary dimension. The formulas for the Green’s function are of interest in their own right. In particular, the explicit representations for a solution to the Dirichlet problem for the biharmonic equation are important in elasticity.