scholarly journals The Dirichlet-problem for harmonic maps from the disk into a lorentzian warped product

1993 ◽  
Vol 10 (2) ◽  
pp. 239-252 ◽  
Author(s):  
Carlo Greco
Keyword(s):  
Dirichlet Problem ◽  
Warped Product ◽  
Harmonic Maps ◽  
Lorentzian Warped Product

On the Dirichlet problem for $$p$$ p -harmonic maps I: compact targets

2014 ◽  
Vol 177 (1) ◽  
pp. 307-322 ◽  
Author(s):  
Stefano Pigola ◽  
Giona Veronelli
Keyword(s):  
Dirichlet Problem ◽  
Harmonic Maps

scholarly journals Hermitian Harmonic Maps into Convex Balls

2007 ◽  
Vol 50 (1) ◽  
pp. 113-122 ◽  
Author(s):  
Zhen Yang Li ◽  
Xi Zhang
Keyword(s):  
Dirichlet Problem ◽  
Heat Flow ◽  
Harmonic Maps ◽  
Hermitian Manifold ◽  
Flow Method ◽  
Hermitian Manifolds ◽  
Compact Hermitian Manifold

AbstractIn this paper, we consider Hermitian harmonic maps from Hermitian manifolds into convex balls. We prove that there exist no non-trivial Hermitian harmonic maps from closed Hermitian manifolds into convex balls, and we use the heat flow method to solve the Dirichlet problem for Hermitian harmonic maps when the domain is a compact Hermitian manifold with non-empty boundary.

scholarly journals Sobolev spaces of maps and the Dirichlet problem for harmonic maps

2019 ◽  
Vol 21 (01) ◽  
pp. 1750091
Author(s):  
Stefano Pigola ◽  
Giona Veronelli
Keyword(s):  
Dirichlet Problem ◽  
Distance Function ◽  
Harmonic Maps ◽  
Geodesic Ball ◽  
Convexity Property ◽  
Normal Coordinate ◽  
Sobolev Classes ◽  
Isometric Embeddings ◽  
Sobolev Maps ◽  
Target Manifold

We give a self-contained treatment of the existence of a regular solution to the Dirichlet problem for harmonic maps into a geodesic ball on which the squared distance function from the origin is strictly convex. No curvature assumptions on the target are required. In this route we introduce a new deformation result which permits to glue a suitable Euclidean end to the geodesic ball without violating the convexity property of the distance function from the fixed origin. We also take the occasion to analyze the relationships between different notions of Sobolev maps when the target manifold is covered by a single normal coordinate chart. In particular, we provide full details on the equivalence between the notions of traced Sobolev classes of bounded maps defined intrinsically and in terms of Euclidean isometric embeddings.

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