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.

Some new properties of Sobolev mappings: intersection theoretical approach

1997 ◽  
Vol 127 (2) ◽  
pp. 337-358 ◽  
Author(s):  
Takeshi Isobe
Keyword(s):  
Sobolev Spaces ◽  
Theoretical Approach ◽  
Energy Gap ◽  
Smooth Maps ◽  
Sobolev Mappings ◽  
Sobolev Maps

In this paper, we give some new examples of the energy gap phenomenon for functionals defined in Sobolev spaces. Our result is independent of that of Giaquinta, Modica and Soucek. We also give some new characterisations of Sobolev maps which can be approximated by smooth 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 Cohomological Criterion for Density of Smooth Maps in Sobolev Spaces Between Two Manifolds

Nematics ◽  
1991 ◽  
pp. 15-23 ◽  
Author(s):  
F. Bethuel ◽  
J. M. Coron ◽  
F. Demengel ◽  
F. Helein
Keyword(s):  
Sobolev Spaces ◽  
Smooth Maps

scholarly journals GAUGE THEORY OF FADDEEV–SKYRME FUNCTIONALS

2010 ◽  
Vol 12 (05) ◽  
pp. 871-908
Author(s):  
SERGIY KOSHKIN
Keyword(s):  
Homotopy Class ◽  
Homogeneous Spaces ◽  
Variational Problems ◽  
Finite Energy ◽  
Energy Functional ◽  
Target Space ◽  
Homotopy Classes ◽  
Existence Of Minimizers ◽  
Sobolev Maps ◽  
Geometric Variational Problems

We study geometric variational problems for a class of nonlinear σ-models in quantum field theory. Mathematically, one needs to minimize an energy functional on homotopy classes of maps from closed 3-manifolds into compact homogeneous spaces G/H. The minimizers are known as Hopfions and exhibit localized knot-like structure. Our main results include proving existence of Hopfions as finite energy Sobolev maps in each (generalized) homotopy class when the target space is a symmetric space. For more general spaces, we obtain a weaker result on existence of minimizers in each 2-homotopy class.Our approach is based on representing maps into G/H by equivalence classes of flat connections. The equivalence is given by gauge symmetry on pullbacks of G → G/H bundles. We work out a gauge calculus for connections under this symmetry, and use it to eliminate non-compactness from the minimization problem by fixing the gauge.

EXISTENCE OF FINITE ENERGY SOLUTIONS FOR ELLIPTIC SYSTEMS WITH L1-VALUED NONLINEARITIES

2008 ◽  
Vol 18 (05) ◽  
pp. 669-687 ◽  
Author(s):  
LUCIO BOCCARDO ◽  
LUIGI ORSINA ◽  
ALESSIO PORRETTA
Keyword(s):  
Elliptic System ◽  
Sobolev Spaces ◽  
Open Subset ◽  
Existence Of Solutions ◽  
Elliptic Systems ◽  
Finite Energy ◽  
Carathéodory Functions ◽  
Bounded Open Subset ◽  
Existence And Regularity Results ◽  
Regularity Results

In this paper, we are going to study the following elliptic system: [Formula: see text] where Ω is a bounded open subset of ℝN, a(x, s) and b(x, s) are positive and coercive Carathéodory functions, and f ∈ LM(Ω). The main purpose of this paper is to prove existence and regularity results with an improved regularity of the function z in the class of Sobolev spaces, and the existence of solutions (u, z) both with finite energy.

Bonded Force Constant Derivation of Lysine-Arginine Cross-linked Advanced Glycation End-Products

2018 ◽  
Author(s):  
Anthony Nash ◽  
Nora H de Leeuw ◽  
Helen L Birch
Keyword(s):  
Molecular Dynamics ◽  
Force Constant ◽  
Computational Study ◽  
Force Constants ◽  
Quantum Mechanical ◽  
Advanced Glycation ◽  
Partial Charges ◽  
Cross Links ◽  
Complete Set ◽  
Dynamics Simulations

<div> <div> <div> <p>The computational study of advanced glycation end-product cross- links remains largely unexplored given the limited availability of bonded force constants and equilibrium values for molecular dynamics force fields. In this article, we present the bonded force constants, atomic partial charges and equilibrium values of the arginine-lysine cross-links DOGDIC, GODIC and MODIC. The Hessian was derived from a series of <i>ab initio</i> quantum mechanical electronic structure calculations and from which a complete set of force constant and equilibrium values were generated using our publicly available software, ForceGen. Short <i>in vacuo</i> molecular dynamics simulations were performed to validate their implementation against quantum mechanical frequency calculations. </p> </div> </div> </div>

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