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Sobolev maps with integer degree and applications to Skyrme’s problem. (English) Zbl 0757.49010
Skyrme’s problem is to minimize the energy \[ E(\varphi)=\int_{\mathbb{R}^ 3}|\nabla\varphi|^ 2+\sum^ 3_{\alpha,\beta=1} \left|{\partial\varphi\over\partial x^ \alpha}\bigwedge{\partial\varphi\over\partial x^ \beta}\right|^ 2dx \] over all maps with finite energy, image in the unit sphere \(S^ 3\), and given degree \[ d(\varphi)={1\over 2\pi^ 2}\int_{\mathbb{R}^ 3}\text{det}(\varphi,\nabla\varphi)dx. \] Previous existence results for this problem have been posed in smaller function spaces because the methods involved use that the degree must be an integer. This paper shows that the degree is an integer as long as the energy is finite, and hence that Skyrme’s problem is always solvable.

49J40 Variational inequalities
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