Circulation integrals and critical Sobolev spaces: problems of optimal constants

Brezis, Haim [UCL] Van Schaftingen, Jean [UCL]

We study various questions related to the best constants in the following inequalities established in [Bourgain-Brezis, 2003; Bourgain-Brezis, 2004; Bourgain-Brezis, 2007]; \[ \Bigl| \int_\Gamma \vec{\varphi} \cdot \vec{t}\, \Bigr|\le C_n \|\nabla \varphi\|_{\mathrm{L}^n} |\Gamma|\;, \] and \[ \Bigl|\int_{\R^n}\vec{\varphi} \cdot \vec{\mu}\, \Bigr|\le C_n \|\nabla \varphi\|_{\mathrm{L}^n} \|\vec \mu\|\;, \] where $\Gamma$ is a closed curve in $\R^n, \vec\varphi \in C^\infty_c (\R^n; \R^n)$ and $\vec \mu$ is a bounded measure on $\R^n$ with values into $\R^n$ such that $\Div \vec\mu = 0$. In 2d the answers are rather complete and closely related to the best constants for Sobolev and isoperimetric inequalities.