Sobolev Maps to the Circle

2021 ◽  
Author(s):  
Haim Brezis ◽  
Petru Mironescu
Keyword(s):  
Sobolev Maps

Sobolev maps with integer degree and applications to Skyrme’s problem

1992 ◽  
Vol 436 (1896) ◽  
pp. 197-201 ◽  
Keyword(s):  
Finite Energy ◽  
Natural Class ◽  
Sobolev Maps

Let ɸ : R 3 → S 3 ⊂ R 4 , ∣ A ( ɸ )∣ 2 ═ Ʃ 3 α,β═1 │∂ ɸ /∂ x α ∧ ∂ ɸ /∂ x β ∣ 2 and let k ϵ Z . Skyrme's problem consists in minimizing the energy ε( ɸ ) : ═ ∫ R 3 ∣∇ ɸ ∣ 2 + ∣ A ( ɸ )∣ 2 d x among maps with degree k ═ d ( ɸ ) : ═ 1/2π 2 ∫ R 3 det ( ɸ , ∇ ɸ ) d x . We show that for all ɸ with finite energy d ( ɸ ) is an integer and then obtain existence of a minimizer of ε in the natural class of maps with finite energy.

A fixed point theorem for a class of Sobolev maps

2006 ◽  
Vol 334 (4) ◽  
pp. 775-782
Author(s):  
Christoph Hamburger
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Sobolev Maps

scholarly journals THE PONTRJAGIN–HOPF INVARIANTS FOR SOBOLEV MAPS

2010 ◽  
Vol 12 (01) ◽  
pp. 121-181 ◽  
Author(s):  
DAVE AUCKLY ◽  
LEV KAPITANSKI
Keyword(s):  
Sobolev Spaces ◽  
Finite Energy ◽  
Homotopy Classification ◽  
Smooth Maps ◽  
Sobolev Maps ◽  
Complete Set ◽  
Integral Degree ◽  
Energy Maps ◽  
Faddeev Model ◽  
Homotopy Invariants

Subtle issues arise when extending homotopy invariants to spaces of functions having little regularity, e.g., Sobolev spaces containing discontinuous functions. Sometimes it is not possible to extend the invariant at all, and sometimes, even when the formulas defining the invariants make sense, they may not have expected properties (e.g., there are maps having non-integral degree).In this paper, we define a complete set of homotopy invariants for maps from three-manifolds to the two-sphere and show that these invariants extend to finite Faddeev energy maps and maps in suitable Sobolev spaces. For smooth maps, our description is proved to be equivalent to Pontrjagin's original homotopy classification from the 1930's. We further show that for the finite energy maps the invariants take on exactly the same values as for smooth maps. We include applications to the Faddeev model.The techniques that we use would also apply to many more problems and/or other functionals. We have tried to make the paper accessible to analysts, geometers and mathematical physicists.

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.

scholarly journals The closing lemma and the planar general density theorem for Sobolev maps

2021 ◽  
Vol 149 (4) ◽  
pp. 1687-1696
Author(s):  
Assis Azevedo ◽  
Davide Azevedo ◽  
Mário Bessa ◽  
Maria Joana Torres
Keyword(s):  
Density Theorem ◽  
Closing Lemma ◽  
Sobolev Maps

scholarly journals Korevaar–Schoen’s directional energy and Ambrosio’s regular Lagrangian flows

2020 ◽  
Author(s):  
Nicola Gigli ◽  
Alexander Tyulenev
Keyword(s):  
Vector Field ◽  
Stability Result ◽  
Sobolev Maps ◽  
Smooth Category ◽  
Euclidean Setting ◽  
Lagrangian Flow ◽  
Do So

Abstract We develop Korevaar–Schoen’s theory of directional energies for metric-valued Sobolev maps in the case of $${\textsf {RCD}}$$ RCD source spaces; to do so we crucially rely on Ambrosio’s concept of Regular Lagrangian Flow. Our review of Korevaar–Schoen’s spaces brings new (even in the smooth category) insights on some aspects of the theory, in particular concerning the notion of ‘differential of a map along a vector field’ and about the parallelogram identity for $${\textsf {CAT}}(0)$$ CAT ( 0 ) targets. To achieve these, one of the ingredients we use is a new (even in the Euclidean setting) stability result for Regular Lagrangian Flows.

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