Brezis–Gallouët inequality

In mathematical analysis, the Brezis–Gallouët inequality, named after Haïm Brezis and Thierry Gallouët, is an inequality valid in 2 spatial dimensions. It shows that a function of two variables which is sufficiently smooth is (essentially) bounded, and provides an explicit bound, which depends only logarithmically on the second derivatives. It is useful in the study of partial differential equations.

Let $$\Omega\subset\mathbb{R}^2$$ be the exterior or the interior of a bounded domain with regular boundary, or $$\mathbb{R}^2$$ itself. Then the Brezis–Gallouët inequality states that there exists a real $$C$$ only depending on $$\Omega$$ such that, for all $$u\in H^2(\Omega)$$ which is not  a.e. equal to 0,
 * $$\displaystyle \|u\|_{L^\infty(\Omega)}\leq C \|u\|_{H^1(\Omega)}\left(1+\Bigl(\log\bigl( 1+\frac{\|u\|_{H^2(\Omega)}}{\|u\|_{H^1(\Omega)}}\bigr)\Bigr)^{1/2}\right).$$

$$ Noticing that, for any $$v\in H^2(\mathbb{R}^2)$$, there holds
 * $$\int_{\mathbb{R}^2} \bigl( (\partial^2_{11} v)^2 + 2(\partial^2_{12} v)^2 + (\partial^2_{22} v)^2\bigr) = \int_{\mathbb{R}^2} \bigl(\partial^2_{11} v+\partial^2_{22} v\bigr)^2,$$

one deduces from the Brezis-Gallouet inequality that there exists $$C>0$$ only depending on $$\Omega$$ such that, for all $$u\in H^2(\Omega)$$ which is not  a.e. equal to 0,
 * $$\displaystyle \|u\|_{L^\infty(\Omega)}\leq C \|u\|_{H^1(\Omega)}\left(1+\Bigl(\log\bigl( 1+\frac{\|\Delta u\|_{L^2(\Omega)}}{\|u\|_{H^1(\Omega)}}\bigr)\Bigr)^{1/2}\right).$$

The previous inequality is close to the way that the Brezis-Gallouet inequality is cited in.