Regularity of minimizing extrinsic polyharmonic maps in the critical dimension

Regularity of extrinsic polyharmonic maps is proved with a method relying on the gradient flow for the underlying functional. This technique permits more general functionals than studied previously, but is restricted to energy minimizers.     

Neck analysis for biharmonic maps

In this paper, we study the blow up of a sequence of (both extrinsic and intrinsic) biharmonic maps in dimension four with bounded energy and show that there is no neck in this process. Moreover, we apply the method to provide new proofs to the removable singularity theorem and energy identity theorem of biharmonic maps.     

On polyharmonic maps into spheres in the critical dimension

We prove that every polyharmonic map u∈Wm,2(Bn,SN−1)$u\in W^{m,2}({\mathbb{B}}^n,{\mathbb{S}}^{N-1})$ is smooth in the critical dimension n=2m$n=2m$. Moreover, in every dimension n  , a weak limit u∈Wm,2(Bn,SN−1)$u\in W^{m,2}({\mathbb{B}}^n,{\mathbb{S}}^{N-1})$ of a sequence of polyharmonic maps uj∈Wm,2(Bn,SN−1)$u_j\in W^{m,2}({\mathbb{B}}^n,{\mathbb{S}}^{N-1})$ is also polyharmonic.     

A regularity result for polyharmonic maps with higher integrability

We prove a regularity result for critical points of the polyharmonic energy in with and p > 1. Our proof is based on a Gagliardo–Nirenberg-type estimate and avoids the moving frame technique. In view of the monotonicity formulae for stationary harmonic and biharmonic maps, we infer partial regularity in theses cases.     

Bubbling analysis for extrinsic biharmonic maps from general Riemannian 4-manifolds

In this paper, we study the blow-up analysis of extrinsic biharmonic maps from a general Riemannian4-manifold into a compact Riemannian manifold. We prove that the energy identity and the no neck property hold with the aid of a Pohozaev identity over general Riemannian manifolds.     

On polyharmonic interpolation

In the present paper we will introduce a new approach to multivariate interpolation by employing polyharmonic functions as interpolants, i.e. by solutions of higher order elliptic equations. We assume that the data arise from C^∞ or analytic functions in the ball BR. We prove two main results on the interpolation of C^∞ or analytic functions f in the ball BR by polyharmonic functions h of a given order of polyharmonicity p.     

Isolated singularities of polyharmonic inequalities

We study nonnegative classical solutions u of the polyharmonic inequality

     

Reproducing kernels for polyharmonic polynomials

The reproducing kernel of the space of all homogeneous polynomials of degree k and polyharmonic order m is computed explicitly, solving a question of A. Fryant and M. K. Vemuri. Received: 17 May 2007, Revised: 27 March 2008     

A note on polyharmonic functions

In this note we prove the following theorem:Suppose 0<p<∞ and α>−1. Then there is a constant C=C(p,m,n,α) such that

     

Removable singularity of the polyharmonic equation

Let x₀∈Ω⊂Rⁿ, n≥2, be a domain and let m≥2. We will prove that a solution u of the polyharmonic equation Δ^mu=0 in Ω?{x₀} has a removable singularity at x₀ if and only if as |x−x₀|→0 for n≥3 and as |x−x₀|→0 for n=2. For m≥2 we will also prove that u has a removable singularity at x₀ if |u(x)|=o(|x−x₀|^2m−n) as |x−x₀|→0 for n≥3 and |u(x)|=o(|x−x₀|^2m−2log(|x−x₀|⁻¹)) as |x−x₀|→0 for n=2.     

Block maps and Fourier analysis

We introduce block maps for subfactors and study their dynamic systems. We prove that the limit points of the dynamic system are positive multiples of biprojections and zero. For the Z_2 case, the asymptotic phenomenon of the block map coincides with that of the 2 D Ising model. The study of block maps requires a further development of our recent work on the Fourier analysis of subfactors. We generalize the notion of sum set estimates in additive combinatorics for subfactors and prove the exact inverse sum set theorem. Using this new method, we characterize the extremal pairs of Young's inequality for subfactors, as well as the extremal operators of the Hausdorff-Young inequality.     

Hile-yeh estimates of eigenvalues of polyharmonic operators

The new bounds of eigenvalues of polyharmonic operators are deduced. This work is a generalization of some kind of estimates of the biharmonic operator. Project supported by the Fund of NNSC.     

A critical elliptic problem for polyharmonic operators

In this paper, we study the existence of solutions for a critical elliptic problem for polyharmonic operators. We prove the existence result in some general domain by minimizing on some infinite-dimensional Finsler manifold for some suitable perturbation of the critical nonlinearity when the dimension of domain is larger than critical one. For the critical dimensions, we prove also the existence of solutions in domains perforated with the small holes. Some unstable solutions are obtained at higher level sets by Coron's topological method, provided that the minimizing solution does not exist.     

A Dirichlet problem for polyharmonic functions

In this article, the Dirichlet problem of polyharmonic functions is considered. As well the explicit expression of the unique solution to the simple Dirichlet problem for polyharmonic functions is obtained by using the decomposition of polyharmonic functions and turning the problem into an equivalent Riemann boundary value problem for polyanalytic functions, as the approach to find the kernel functions of the solution for the general Dirichlet problem is given. Project supported by NNSF of China.     

On polyharmonic helices in space forms

Archiv der Mathematik - In this article, we study polyharmonic curves of constant curvature where we mostly focus on the case of curves on the sphere. We classify polyharmonic curves of constant...