调和映射中的弱单调不等式和部分正则性

对弱调和映射引入局部弱单调不等式的概念 ,并得到了这类弱调和映射的一些结果 ,如局部弱单调不等式几乎等价于ε-正则性等 .     

靶流形为齐次流形的弱次椭圆Q调和映射的正则性

该文证明了靶流形为齐次流形的弱次椭圆Q调和映射是内部正则的，这里Q是定义域的 齐次维数。这一结果推广了Hajlasz和Strzelecki的相应结果[2].作为推论得到了靶流形为齐次流形的p维p调和映射的正则性.     

非齐次p-调和映射方程组弱驻点解的正则性

讨论非齐次p－调和映射方程组弱驻点解的内部正则性，证明了弱驻点解满足拟单调不等式，得到了弱驻点解梯度的Holder连续性。     

弱P-调和映射流的紧性性质

分别考虑了映入球面及紧致的齐性Ｒｉｅｍａｎｎｉａｎ空间的弱Ｐ－调和映射流；通过球面及齐性Ｒｉｅｍａｎｎｉａｎ空间的对称性质，证明了弱Ｐ－调和映射流的紧性性质．     

退化（k1，k2）拟正则映射的充分条件

研究了退化弱（k1,k2）拟正则映射的正则性．利用Holder不等式、Sobolev空间的空间分析方法,以及内插定理等工具,给出了退化弱（k1,k2）拟正则映射事实上为退化（k1,k2）拟正则映射的一个充分条件,其结果对非退化情形也成立．     

退化(K_1,K_2)拟正则映射的充分条件

研究了退化弱(k1,k2)拟正则映射的正则性.利用H lder不等式、Sobolev空间的空间分析方法,以及内插定理等工具,给出了退化弱(k1,k2)拟正则映射事实上为退化(k1,k2)拟正则映射的一个充分条件,其结果对非退化情形也成立.     

关于弱－θ－加细空间的闭L原象

本文证明了，当定义域空间是正则空间时，弱－θ－加细性在闭Ｌ映射下是逆保持的，并给出例子说明，去掉定义域空间的正则性，上述结论不成立。     

关于弱－θ－加细空间的闭L原象

本文证明了，当定义域空间是正则空间时，弱θ＾－－加细性在闭Ｌ映射下是逆保持的，并给出例子说明，去掉定义域空正则性，上述结论不成立。     

序列式MESO紧空间的闭Lindelof逆象

证明了正则空间中闭Lindelof映射逆保持序列式meso紧性,从而改进了Mancuso V J关于正则空间中完备映射逆保持meso紧性这一结果;进一步我们指出定理条件中原象空间的正则性不可被省略而象空间的正则性可以用原象空间的正规性来替代.     

带自由边界的调和映射存在性

本文研究带自由边界的调和映射的存在性.用Sack-Uhlenbeck的挠动Morse理论及Micallef和Moore的Morse指标的估计方法,我们得到了从圆盘到Riemann流形的带自由边界的调和映射的存在性.     

A regularity theory for intrinsic minimising fractional harmonic maps

We define and develop an interior partial regularity theory for intrinsic energy minimising fractional harmonic maps from Euclidean space into smooth compact Riemannian manifolds for fractional powers strictly between zero and one. Intrinsic fractional harmonic maps are critical points of an energy whose first variation is a Dirichlet to Neumann map for the harmonic map problem on a half-space with a Riemannian metric which can degenerate/become singular along the boundary, depending on the fractional power. Similarly to the approach used to prove regularity for stationary intrinsic semi-harmonic maps, we take advantage of the connection between fractional harmonic maps and free boundary problems for harmonic maps in order to develop a partial regularity theory for the fractional harmonic maps we consider. In particular, we prove partial regularity for locally minimising harmonic maps with (partially) free boundary data on half-spaces with the aforementioned metrics up to the boundary; fractional harmonic maps then inherit this regularity. As a by-product of our methods we shed some new light on the monotonicity of the average energy of solutions of the degenerate linear elliptic equation related to fractional harmonic functions.     

Concentrated boundary data and axially symmetric harmonic maps

We examine the existence problem for harmonic maps between the three-dimensional ball and the two-sphere. We utilize results on the classification of harmonic maps into hemispheres and a result on the regularity of the weak limit of energy minimizers over the class of axially symmetric maps to establish the existence of asmooth harmonic extension for boundary data suitably “concentrated” away from the axis of symmetry. In addition, we establish convergence results for the harmonic map heat flow problem for suitably “concentrated” axially symmetric initial and boundary data.     

Regularity of harmonic maps into a static Lorentzian manifold

We study the regularity of harmonic maps from Riemannian manifold into a static Lorentzian manifold. We show that when the domain manifold is two-dimensional, any weakly harmonic map is smooth. We also show that when dimension n of the domain manifold is greater than two, there exists a weakly harmonic map for the Dirichlet problem which is smooth except for a closed set whose (n − 2)-dimensional Hausdorff measure is zero.     

A boundary monotonicity inequality for variationally biharmonic maps and applications to regularity theory

We derive a boundary monotonicity formula for a class of biharmonic maps with Dirichlet boundary conditions. A monotonicity formula is crucial in the theory of partial regularity in supercritical dimensions. As a consequence of such a boundary monotonicity formula, one is able to show partial regularity for variationally biharmonic maps and full boundary regularity for minimizing biharmonic maps.     

Harmonic Maps Between Alexandrov Spaces

In this paper, we shall discuss the existence, uniqueness and regularity of harmonic maps from an Alexandrov space into a geodesic space with curvature \(\leqslant 1\) in the sense of Alexandrov.     

Multiple solutions of the Dirichlet problem for harmonic maps from discs into 2-spheres

In this paper we study the topology of the space of maps from the disc into the 2-sphere with given boundary data by comparing to the topology of the space of rational functions. And we prove theorems about multiple solutions of the Dirichlet problem for harmonic maps in one homotopy class.     

A Remark on Isolated Singularities at the Free Boundary of Harmonic Maps

Using a simple lemma on harmonic functions it is shown how well known results can be combined to exclude isolated singularities at the free boundary of two-dimensional weakly harmonic maps.     

Unique solvability of the Dirichlet problem¶for weakly harmonic maps

We prove existence and uniqueness of weakly harmonic maps from the unit ball in ℝ ⁿ (with n≥ 3) to a smooth compact target manifold within the class of maps with small scaled energy for suitable boundary data. Received: 9 June 2000 / Revised version: 17 April 2001     

Boundary regularity for minimizers of the micromagnetic energy functional

Motivated by the construction of time-periodic solutions for the three-dimensional Landau-Lifshitz-Gilbert equation in the case of soft and small ferromagnetic particles, we investigate the regularity properties of minimizers of the micromagnetic energy functional at the boundary. In particular, we show that minimizers are regular provided the volume of the particle is sufficiently small. The approach uses a reflection construction at the boundary and an adaption of the well-known regularity theory for minimizing harmonic maps into spheres.     

The Dirichlet problem for harmonic maps from Riemannian polyhedra to spaces of upper bounded curvature

This is a continuation of the Cambridge Tract ``Harmonic maps between Riemannian polyhedra', by J. Eells and the present author. The variational solution to the Dirichlet problem for harmonic maps with countinuous boundary data is shown to be continuous up to the boundary, and thereby uniquely determined. The domain space is a compact admissible Riemannian polyhedron with boundary, while the target can be, for example, a simply connected complete geodesic space of nonpositive Alexandrov curvature; alternatively, the target may have upper bounded curvature provided that the maps have a suitably small range. Essentially in the former setting it is further shown that a harmonic map pulls convex functions in the target back to subharmonic functions in the domain.