Sobolev spaces and harmonic maps between singular spaces

Giovanni Gregori

doi:10.1007/s005260050096

Sobolev spaces and harmonic maps between singular spaces

Calculus of Variations and Partial Differential Equations ◽

10.1007/s005260050096 ◽

1998 ◽

Vol 7 (1) ◽

pp. 1-18 ◽

Cited By ~ 13

Author(s):

Giovanni Gregori

Keyword(s):

Sobolev Spaces ◽

Harmonic Maps ◽

Singular Spaces

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Harmonic maps into singular spaces andp-adic superrigidity for lattices in groups of rank one

Publications mathématiques de l IHÉS ◽

10.1007/bf02699433 ◽

1992 ◽

Vol 76 (1) ◽

pp. 165-246 ◽

Cited By ~ 125

Author(s):

Mikhail Gromov ◽

Richard Schoen

Keyword(s):

Harmonic Maps ◽

Singular Spaces ◽

Rank One

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UNIQUENESS THEOREMS FOR HARMONIC MAPS INTO METRIC SPACES

Communications in Contemporary Mathematics ◽

10.1142/s0219199702000828 ◽

2002 ◽

Vol 04 (04) ◽

pp. 725-750 ◽

Cited By ~ 2

Author(s):

CHIKAKO MESE

Keyword(s):

Riemannian Manifolds ◽

Metric Space ◽

Metric Spaces ◽

Harmonic Maps ◽

Uniqueness Theorems ◽

Singular Spaces ◽

Domain Space ◽

Recent Developments

Recent developments extend much of the known theory of classical harmonic maps between smooth Riemannian manifolds to the case when the target is a metric space of curvature bounded from above. In particular, the existence and regularity theorems for harmonic maps into these singular spaces have been successfully generalized. Furthermore, the uniqueness of harmonic maps is known when the domain has a boundary (with a smallness of image condition if the target curvature is bounded from above by a positive number). In this paper, we will address the question of uniqueness when the domain space is without a boundary in two cases: one, when the curvature of the target is strictly negative and two, for a map between surfaces with nonpositive target curvature.

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ON THE DEFINITIONS OF SOBOLEV AND BV SPACES INTO SINGULAR SPACES AND THE TRACE PROBLEM

Communications in Contemporary Mathematics ◽

10.1142/s0219199707002502 ◽

2007 ◽

Vol 09 (04) ◽

pp. 473-513 ◽

Cited By ~ 13

Author(s):

DAVID CHIRON

Keyword(s):

Sobolev Spaces ◽

Metric Spaces ◽

The Other ◽

Dirichlet Energy ◽

Singular Spaces ◽

Other Hand ◽

The One

The purpose of this paper is to relate two notions of Sobolev and BV spaces into metric spaces, due to Korevaar and Schoen on the one hand, and Jost on the other hand. We prove that these two notions coincide and define the same p-energies. We review also other definitions, due to Ambrosio (for BV maps into metric spaces), Reshetnyak and finally to the notion of Newtonian–Sobolev spaces. These last approaches define the same Sobolev (or BV) spaces, but with a different energy, which does not extend the standard Dirichlet energy. We also prove a characterization of Sobolev spaces in the spirit of Bourgain, Brezis and Mironescu in terms of "limit" of the space Ws,p as s → 1, 0 < s < 1, and finally following the approach proposed by Nguyen. We also establish the [Formula: see text] regularity of traces of maps in Ws,p (0 < s ≤ 1 < sp).

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