Circulation integrals and critical Sobolev spaces: problems of optimal constants
(2008) Perspectives in Partial Differential Equations, Harmonic Analysis and Applications — ISBN: [978-0-8218-4424-3], p. 33-47, published
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- Abstract
- 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.
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Brezis, H., & Van Schaftingen, J. (2008). Circulation integrals and critical Sobolev spaces: problems of optimal constants. In Dorina Mitrea (ed.), Perspectives in Partial Differential Equations, Harmonic Analysis and Applications (p. p. 33-47). American Mathematical Society. https://hdl.handle.net/2078.5/246365