scholarly journals Sobolev spaces and harmonic maps for metric space targets

1993 ◽  
Vol 1 (4) ◽  
pp. 561-659 ◽  
Author(s):  
Nicholas J. Korevaar ◽  
Richard M. Schoen
Keyword(s):  
Sobolev Spaces ◽  
Metric Space ◽  
Harmonic Maps

scholarly journals UNIQUENESS THEOREMS FOR HARMONIC MAPS INTO METRIC SPACES

2002 ◽  
Vol 04 (04) ◽  
pp. 725-750 ◽  
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.

scholarly journals Traces of Besov, Triebel-Lizorkin and Sobolev Spaces on Metric Spaces

2017 ◽  
Vol 5 (1) ◽  
pp. 98-115 ◽  
Author(s):  
Eero Saksman ◽  
Tomás Soto
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Metric Space ◽  
Besov Spaces ◽  
Metric Spaces ◽  
Function Spaces ◽  
First Order ◽  
Trace Theorems ◽  
Euclidean Setting ◽  
Regular Subset

Abstract We establish trace theorems for function spaces defined on general Ahlfors regular metric spaces Z. The results cover the Triebel-Lizorkin spaces and the Besov spaces for smoothness indices s < 1, as well as the first order Hajłasz-Sobolev space M1,p(Z). They generalize the classical results from the Euclidean setting, since the traces of these function spaces onto any closed Ahlfors regular subset F ⊂ Z are Besov spaces defined intrinsically on F. Our method employs the definitions of the function spaces via hyperbolic fillings of the underlying metric space.

Generalized multivalued integral type contraction on weak partial metric space

2020 ◽  
Vol 13 (3) ◽  
pp. 145-151
Author(s):  
Öztürk zlem Acar ◽  
Sümeyye Coşkun
Keyword(s):  
Metric Space ◽  
Partial Metric Space ◽  
Partial Metric ◽  
Integral Type ◽  
Type Contraction

On Asymmetric Distances

2013 ◽  
Vol 1 ◽  
pp. 200-231 ◽  
Author(s):  
Andrea C.G. Mennucci
Keyword(s):  
Total Variation ◽  
Calculus Of Variations ◽  
Distance Function ◽  
Metric Space ◽  
Metric Spaces ◽  
Geodesic Segment ◽  
Intrinsic Metric ◽  
Typical Application ◽  
Variation Formula

Abstract In this paper we discuss asymmetric length structures and asymmetric metric spaces. A length structure induces a (semi)distance function; by using the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempotent. As a typical application, we consider the length of paths defined by a Finslerian functional in Calculus of Variations. In the second part we generalize the setting of General metric spaces of Busemann, and discuss the newly found aspects of the theory: we identify three interesting classes of paths, and compare them; we note that a geodesic segment (as defined by Busemann) is not necessarily continuous in our setting; hence we present three different notions of intrinsic metric space.

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