Dirichlet's boundary value problem for harmonic mappings of Riemannian manifolds

1976 ◽  
Vol 147 (3) ◽  
pp. 225-236 ◽  
Author(s):  
Stefan Hildebrandt ◽  
Helmut Kaul ◽  
Kjell-Ove Widman
Keyword(s):  
Boundary Value Problem ◽  
Riemannian Manifolds ◽  
Boundary Value ◽  
Harmonic Mappings

Stability and constant boundary-value problems of harmonic maps with potential

2000 ◽  
Vol 68 (2) ◽  
pp. 145-154 ◽  
Author(s):  
Qun Chen
Keyword(s):  
Riemannian Manifold ◽  
Energy Density ◽  
Boundary Value Problem ◽  
Riemannian Manifolds ◽  
General Class ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Boundary Value ◽  
The Stability ◽  
Harmonic Maps With Potential

AbstractLet M, N be Riemannian manifolds, f: M → N a harmonic map with potential H, namely, a smooth critical point of the functional EH(f) = ∫M[e(f)−H(f)], where e(f) is the energy density of f. Some results concerning the stability of these maps between spheres and any Riemannian manifold are given. For a general class of M, this paper also gives a result on the constant boundary-value problem which generalizes the result of Karcher-Wood even in the case of the usual harmonic maps. It can also be applied to the static Landau-Lifshitz equations.

scholarly journals On the two-point boundary value problem for quadratic second-order differential equations and inclusions on manifolds

2006 ◽  
Vol 2006 ◽  
pp. 1-9 ◽  
Author(s):  
Yuri E. Gliklikh ◽  
Peter S. Zykov
Keyword(s):  
Boundary Value Problem ◽  
Riemannian Manifolds ◽  
Boundary Value ◽  
Second Order ◽  
Quadratic Growth ◽  
Point Boundary ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Differential Equations And Inclusions ◽  
Complete Riemannian Manifolds

The two-point boundary value problem for second-order differential inclusions of the form(D/dt)m˙(t)∈F(t,m(t),m˙(t))on complete Riemannian manifolds is investigated for a couple of points, nonconjugate along at least one geodesic of Levi-Civitá connection, whereD/dtis the covariant derivative of Levi-Civitá connection andF(t,m,X)is a set-valued vector with quadratic or less than quadratic growth in the third argument. Some interrelations between certain geometric characteristics, the distance between points, and the norm of right-hand side are found that guarantee solvability of the above problem forFwith quadratic growth inX. It is shown that this interrelation holds for all inclusions withFhaving less than quadratic growth inX, and so for them the problem is solvable.

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