The limiting Dirac-Sobolev inequality
Résumé
We prove the critical Dirac-Sobolev inequality for $p\in(1,3)$. It follows that the Dirac Sobolev spaces are equivalent to classical Sobolev spaces if and only if $p\in(1,3)$. We prove the cocompactness of $L^{p^{*}}(\mathbb{R}^{3})$ in }{\normalsize $\dot{\mathbf{H}}^{1,p}(\mathbb{R}^{3})$. As an application, we prove the existence of minimizers to a class of isoperimetric problems.