## Abstract

Let [Note: See the image of page 197 for this formatted text] $\phi $:R $^{3}\rightarrow $ S$^{3}\subset R^{4}$, $\quad|$A($\phi $)$|^{2}$ = $\sum_{\alpha,\beta =1}^{3}\left|\frac{\partial \phi}{\partial x_{\alpha}}\wedge \frac{\partial \phi}{\partial x_{\beta}}\right|^{2}$ and let k $\in $ Z. Skyrme's problem consists in minimizing the energy [Note: See the image of page 197 for this formatted text] $\scr{E}$($\phi $): = $\int_{R^{3}}|\nabla \phi|^{2}$+ $|$A($\phi $)$|^{2}$ dx among maps with degree [Note: See the image of page 197 for this formatted text] k = d($\phi $): = $\frac{1}{2\pi ^{2}}\int_{R^{3}}$ det ($\phi $, $\nabla \phi $) dx. We show that for all $\phi $ with finite energy d($\phi $) is an integer and then obtain existence of a minimizer of $\scr{E}$ in the natural class of maps with finite energy.

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