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.
PIGOLA, STEFANO (Corresponding)
|Data di pubblicazione:||9999|
|Titolo:||Sobolev spaces of maps and the Dirichlet problem for harmonic maps|
|Rivista:||COMMUNICATIONS IN CONTEMPORARY MATHEMATICS|
|Digital Object Identifier (DOI):||10.1142/S0219199717500912|
|Parole Chiave:||Harmonic maps; Dirichlet problem; Sobolev spaces; convex exhaustions; gluing flat ends|
|Appare nelle tipologie:||Articolo su Rivista|