A bifurcation result for harmonic maps from an annulus to with not symmetric boundary data

We consider the problem of minimizing the energy of the maps from the annulus to such that is equal to for , and to , for , where is a fixed angle.

We prove that the minimum is attained at a unique harmonic map which is a planar map if , while it is not planar in the case \pi^2$">.

Moreover, we show that tends to as , where minimizes the energy of the maps from to , with the boundary condition , .

     

The existence of harmonic maps from an annulus toS 2 with symmetric boundary value

In this paper, we use the deformation method andG-equivariant theory to prove the existence and multiplicity of harmonic maps from an annulus to the unit sphere inℝ ³ with symmetric boundary value. In particular, we can get infinitely many homotopically different harmonic maps if the boundary value isS ¹-equivariant and nonconstant. This research partially supported by the NNSF, P.R. China.     

Axially symmetric harmonic maps minimizing a relaxed energy

Here we obtain various results on the class of axially symmetric harmonic maps from B ³ to S ². We find some new classes of non-minimizing harmonic maps exhibiting unusual singular behavior. Optimal partial regularity estimates are obtained for mappings which minimize, among axially symmetric maps, various relaxed energies which have been studied in [4]and [11].     

Twistor fibrations over Hermitian symmetric spaces and harmonic maps

Given a twistor space over a Hermitian symmetric space of compact type we construct a map onto a twistor space over another inner symmetric space of compact type. This map is holomorphic and preserves the superhorizontal distributions. We describe an application to harmonic maps.     

Propagation of high-frequency harmonic elastic waves from an axially symmetric cavity

An asymptotic ray method is developed for studying propagation of high-frequency harmonic elastic waves from a smooth, convex, axially symmetric cavity. The dynamic stresses near a prolate spheroidal cavity, to whose surface is applied a normal, angle-independent load, are investigated as a specific numerical example. It is shown that account of three terms of the ray series makes it possible to find quite accurate solutions even when the wave length is comparable with the cavity sizes.Translated from Teorr eticheskaya i Prikladnaya Mekhanika, No. 18, pp. 87–95, 1987.     

Harmonic map heat flow for axially symmetric data

We examine the harmonic map heat flow problem for maps between the three-dimensional ball and the two-sphere. We give blow-up results for certain initial data. We establish convergence results for suitable axially symmetric initial data, and discuss generalizations to higher dimensions.     

An initial value approach to rotationally symmetric harmonic maps

We study the effect of the varying y′(0) on the existence and asymptotic behavior of solutions for the initial value problem

     

Interior and boundary monotonicity formulas for stationary harmonic maps

We derive monotonicity formulas for stationary harmonic maps. The proofs are given in coordinates.     

Fully nonlinear curvature flow of axially symmetric hypersurfaces with boundary conditions

Inspired by earlier results on the quasilinear mean curvature flow, and recent investigations of fully nonlinear curvature flow of closed hypersurfaces which are not convex, we consider contraction of axially symmetric hypersurfaces by convex, degree-one homogeneous fully nonlinear functions of curvature. With a natural class of Neumann boundary conditions, we show that evolving hypersurfaces exist for a finite maximal time. The maximal time is characterised by a curvature singularity at either boundary. Some results continue to hold in the cases of mixed Neumann–Dirichlet boundary conditions and more general curvature-dependent speeds.     

A boundary value problem for Hermitian harmonic maps and applications

We study the existence and uniqueness problems for Hermitian harmonic maps from Hermitian manifolds with boundary to Riemannian manifolds of nonpositive sectional curvature and with convex boundary. The complex analyticity of such maps and the related rigidity problems are also investigated.

     

Variationally harmonic maps with general boundary conditions: Boundary regularity

Let and be Riemannian manifolds, compact without boundary. We develop a definition of a variationally harmonic map with respect to a general boundary condition of the kind u(x)∊Γ(x) for a.e. , where are given submanifolds depending smoothly on x. The given definition of variationally harmonic maps is slightly more restrictive, but also more natural than the usual definition of stationary harmonic maps. After deducing an energy monotonicity formula, it is possible to derive a regularity theory for variationally harmonic maps with general boundary data. The results include full boundary regularity in the Dirichlet boundary case Γ(x) = {g(x)} for if does not carry a nonconstant harmonic 2-sphere.     

Partial regularity for stationary harmonic maps at a free boundary

Let and be smooth Riemannian manifolds, of the dimension n≥2 with nonempty boundary, and compact without boundary. We consider stationary harmonic maps u ∈ H¹(, ) with a free boundary condition of the type u(∂) ⊂ Γ, given a submanifold Γ⊂. We prove partial boundary regularity, namely (sing(u))=0, a result that was until now only known in the interior of the domain (see [B]). The key of the proof is a new lemma that allows an extension of u by a reflection construction. Once the partial regularity theorem is known, it is possible to reduce the dimension of the singular set further under additional assumptions on the target manifold and the submanifold Γ.     

Regularity of harmonic maps from the Minkowski space into rotationally symmetric manifolds

We prove global regularity for the solution to the Cauchy problem with regular data for an equivariant harmonic map from the 2 + 1-dimensional Minkowski space into a two-dimensional, rotationally symmetric, and geodesically convex Riemannian manifold.     

Rotationally symmetric harmonic maps from a ball into a warped product manifold

This paper deals with the existence problem for rotationally symmetric harmonic maps from an Euclidean unit ball B ⁿ or ⁿ into a warped product manifold N_f=[0, r₀)x_fS^n–1.     

Energy minimizing harmonic maps with an obstacle at the free boundary

We consider partial regularity for energy minimizing maps satisfying a partially free boundary condition. This condition takes the form of the requirement that a relatively open subset of the boundary of the domain manifold be mapped into a closed submanifold with non-empty boundary, contained in the target manifold. We obtain an optimal estimate on the Hausdorff dimension of the singular set of such a map. Our result can be interpreted as regularity result for a vector-valued Signorini, or thin-obstacle, problem.