scholarly journals Partial Regularity for Harmonic Maps into Spheres at a Singular or Degenerate Free Boundary

2022 ◽  
Vol 32 (2) ◽  
Author(s):  
Roger Moser ◽  
James Roberts
Keyword(s):  
Free Boundary ◽  
Distance Function ◽  
Boundary Data ◽  
Partial Regularity ◽  
Harmonic Maps

AbstractWe prove partial regularity of weakly stationary harmonic maps with (partially) free boundary data on manifolds where the domain metric may degenerate or become singular along the free boundary at the rate $$d^\alpha $$ d α for the distance function d from the boundary.

A partial regularity theorem for harmonic maps at a free boundary

1989 ◽  
Vol 2 (4) ◽  
pp. 299-343 ◽  
Author(s):  
Frank Duzaar ◽  
Klaus Steffen
Keyword(s):  
Free Boundary ◽  
Partial Regularity ◽  
Harmonic Maps ◽  
Regularity Theorem

scholarly journals Partial regularity of $p(x)$-harmonic maps

2012 ◽  
Vol 365 (6) ◽  
pp. 3329-3353 ◽  
Author(s):  
Maria Alessandra Ragusa ◽  
Atsushi Tachikawa ◽  
Hiroshi Takabayashi
Keyword(s):  
Partial Regularity ◽  
Harmonic Maps

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.

Sign in / Sign up

Export Citation Format

Share Document