# A New Setting For Skyrme’s Problem

• Maria J. Esteban
Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 4)

## Abstract

In [3] we gave some existence results for the general Skyrme’s problem, without any symmetry assumption. This problem arises as a natural model when searching to identify baryons as solitons in meson field theory. For more details about the physics motivation see [1,7,9,10]. It can be defined as follows. A class of functions ø : R 3 → S3, X,has to be defined so that the Skyrme’s energy functional
$$\varepsilon (\Phi ) = \int_{{R^3}} {|\nabla \phi } {|^2} + |A(\phi ){|^2}dx,$$
(1)
$$(with A(\phi ) = \left( {\frac{{\partial \phi }}{{\partial {x_i}}} \wedge \frac{{\partial \phi }}{{\partial {X_j}}}} \right) i,j = 1,2,3)$$ is finitely defined in X. Then ε has to be minimized in the subclass of X formed by the functions ø having a “topological degree” d(ø) equal to k ∈ Z,where
$$d(\phi ) = \frac{1}{{2{\pi ^2}}}\int_{{R^3}} {\det (\phi ,\nabla \phi )} dx.$$
(2)
Physically d(ø) represents the baryonic number and has then to be an integer. So one has to consider a class X in which d takes naturally only integer values. Actually when ø ∈ Cl (R3, S3), d(ø) is the topological degree of ø o E,E being a stereographic projection mapping 5 S3 into R 3. Therefore it is natural to define X as a class of functions which can be approached by smooth functions in a convenient sense. In [3] such a class was chosen where the corresponding density property was quite strong. Then existence results for the Skyrme’s problem in that setting were given. Actually those results are valid only if one proves the approximation of finite energy functions by smooth functions in a rather strong way, and this is an open problem.

## Keywords

Density Property Finite Energy Topological Degree Skyrme Model Duality Product
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