Biharmonic Maps from Tori into a 2-Sphere

Biharmonic maps are generalizations of harmonic maps. A well-known result on harmonic maps between surfaces shows that there exists no harmonic map from a torus into a sphere(whatever the metrics chosen) in the homotopy class of maps of Brower degree±1. It would be interesting to know if there exists any biharmonic map in that homotopy class of maps. The authors obtain some classifications on biharmonic maps from a torus into a sphere, where the torus is provided with a flat or a class of non-flat metrics whilst the sphere is provided with the standard metric. The results in this paper show that there exists no proper biharmonic maps of degree±1 in a large family of maps from a torus into a sphere.     

双调和方程柯西问题的修正的Tikhonov正则化方法

本文研究了双调和方程柯西问题,这类是不适定的,即问题的解(如果存在)不连续依赖于测量数据.首先在精确解的先验假设下给出问题的条件稳定性结果.接着利用修正的Tikhonov正则化方法求解此不适定问题.在先验和后验正则化参数选取规则下,给出正则解和精确解之间的误差估计式.最后给出几个数值例子验证此正则化方法求解此类反问题的有效性.     

Mortar Finite Volume Method with Adini Element for Biharmonic Problem

In this paper, we construct and analyse a mortar finite volume method for the discretization for the biharmonic problem in R2. This method is based on the mortar-type Adini nonconforming finite element spaces. The optimal order H2-seminorm error estimate between the exact solution and the mortar Adini finite volume solution of the biharmonic equation is established.     

Some Results of Biharmonic Maps

In this paper, we investigate biharmonic maps from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. We obtain some non-existence results for these maps.     

Biharmonic homogeneous hypersurfaces in compact symmetric spaces

In this paper, we study biharmonic hypersurfaces in Einstein manifolds. Then, we determine all the biharmonic hypersurfaces in irreducible symmetric spaces of compact type which are regular orbits of commutative Hermann actions of cohomogeneity one.     

具临界位势和权函数的非线性双调和问题

本文考虑具有临界位势和临界权函数的四维非线性重调和问题,证明非平凡解的存在性.     

On Constructing Biharmonic Maps and Metrics

We construct biharmonic nonharmonic maps between Riemannian manifoldsM and N by first making the ansatz that M N be aharmonic map and then deforming the metric conformally on M to render biharmonic. The deformation will, in general, destroy theharmonicity of . We call a metric which renders the identity mapbiharmonic, a biharmonic metric. On an Einstein manifold, theonly conformally equivalent biharmonic metrics are defined byisoparametric functions.     

半线性奇系数临界双调和方程的Dirichlet问题

主要探讨了两类半线性双调和Dirichlet问题：奇系数次临界问题和临界但带较弱奇性问题 ，得出了在临界维数和正常维数不同情况下都至少有一个正解的结论. 同时也研究了临界维数的消失问题，比较了奇系数与较弱奇性不同情况下临界维数的变化，得出奇性越大临界维数越少的结论.      

A General Approach to Equivariant Biharmonic Maps

In this paper we describe a 1-dimensional variational approach to the analytical construction of equivariant biharmonic maps. Our goal is to provide a direct method which enables analysts to compute directly the analytical conditions which guarantee biharmonicity in the presence of suitable symmetries. In the second part of our work, we illustrate and discuss some examples. In particular, we obtain a 1-dimensional stability result, and also show that biharmonic maps do not satisfy the classical maximum principle proved by Sampson for harmonic maps.     

On the Regularity Theories for Harmonic Maps from Finsler Manifolds

The author studies the regularity of energy minimizing maps from Finsler manifolds to Riemannian manifolds.It is also shown that the energy minimizing maps are smooth,when the target manifolds have no ...     

Sobolev Hardy不等式与临界双重调和问题

该文讨论一类带有奇异系数的双重调和方程〖JB({〗△2u-μ[SX(]u[]|x|s[SX)]=f(x,u),\=u=[SX(]u[]ν[SX)]=0,〖JB)〗\ \ 〖JB(〗x∈Ω,x∈Ω,[JB)] 这里ΩRN是包含0的有界光滑区域，u∈H20(Ω)，μ∈R是参数，0≤s≤2，△2=△△表示双重拉普拉斯算子.当f(x,u)=up,p=[SX(]2N[]N-4[SX)]时，上述问题就是一个临界双重调和问题. 该文运用Sobolev Hardy不等式和变分方法，得到它的解的存在性的一些结果.     

Sobolev-Hardy不等式与临界双重调和问题

该文讨论一类带有奇异系数的双重调和方程{△^2u-μu/|x|^s=f(x,u),x∈Ω,u=δu/δv=0,x∈δΩ，这里Ω包含R^N是包含0的有界光滑区域，u∈H0^2(Ω），μ∈R是参数，0≤s≤2,△^2=△△表示双重拉普拉斯算子，当f(x,u)=u^p,p=2N/N-4时，上述问题就是一个临界双重调和问题，该文运用Sobolev-Hardy不等式和变分方法，得到它的解的存在性的一些结果。     

Existence and Multiplicity of Solutions for a Biharmonic Kirchhoff Equation in $\mathbb{R}^5$

We consider the biharmonic equation $\Delta^2u-\left(a+b\int_{\R^5}|\nabla u|^2dx\right)\Delta u\\+V(x)u=f(u)$, where $V(x)$ and $f(u)$ are continuous functions. By using a perturbation approach and the symmetric mountain pass theorem, the existence and multiplicity of solutions for this equation are obtained, and the power-type case $f(u)=|u|^{p-2}u$ is extended to $p\in(2,10)$, where it was assumed $p\in(4,10)$ in many papers.     

一类半线性双调和方程的整体解

本文主要目的是证明一类半线性双调和方程的正的整体解的存在性,推广了T.Kusano和C.A.Swanson的部分结果。     

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.     

Regularity for weakly Dirac-harmonic maps to hypersurfaces

We prove that a weakly Dirac-harmonic map from a Riemann spin surface to a compact hypersurface is smooth. Supported by IMPRS “Mathematics in the Sciences” and the Klaus Tschira Foundation.     

关于上半连续集值映象的连续选择问题

本文进一步研究了上半连续集值映象的连续选择问题．首先,用一反例指出Ａｕｂｉｎ Ｊ、 Ｐ［１］中一结果不能成立．另外,对上半连续映象的σ－可选择性给出了另一逼近形式的结果．     

Localization of Generalized Normal Maps and Stability of Variational Inequalities in Reflexive Banach Spaces

In this paper, we introduce a localized version of generalized normal maps as well as generalized natural mappings. By using these concepts, we study continuity properties of the solution map of parametric variational inequalities in reflexive Banach spaces. This localization permits us to deal with variational conditions posed on sets that may not be convex and to establish existence and continuity of solutions. We also establish homeomorhisms between the solution set of variational inequalities and the solution set of generalized normal maps. Using these homeomorphisms and the degree theory, we show that the solution map of parametric variational inequalities is lower semicontinuous. Our results extend some results of Robinson (Set-Valued Anal 12:259–274, 2004). The authors wish to express their sincere appreciation to Professor Stephen M. Robinson, Department of Industrial and Systems Engineering, University of Wisconsin-Madison, for his valuable comments and suggestions. This research was partially supported by a grant from National Science Council of Taiwan, ROC.