A flow on $ S^2 $ presenting the ball as its minimal set

Abstract

<p style='text-indent:20px;'>The main goal of this paper is to present the existence of a vector field tangent to the unit sphere <inline-formula><tex-math id="M2">\begin{document}$ S^2 $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M3">\begin{document}$ S^2 $\end{document}</tex-math></inline-formula> itself is a minimal set. This is reached using a piecewise smooth (discontinuous) vector field and following the Filippov's convention on the switching manifold. As a consequence, none regularization process applied to the initial model can be topologically equivalent to it and we obtain a vector field tangent to <inline-formula><tex-math id="M4">\begin{document}$ S^2 $\end{document}</tex-math></inline-formula> without equilibria.