Английская Википедия:Brezis–Gallouët inequality

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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 <math>\Omega\subset\mathbb{R}^2</math> be the exterior or the interior of a bounded domain with regular boundary, or <math>\mathbb{R}^2</math> itself. Then the Brezis–Gallouët inequality states that there exists a real <math>C</math> only depending on <math>\Omega</math> such that, for all <math>u\in H^2(\Omega)</math> which is not a.e. equal to 0,

<math>\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).</math>

Шаблон:Math proof {1+R}.</math> The inequality is then proven, in the case <math>\|u\|_{H^1(\Omega)} = 1</math>, by letting <math>R =\|u\|_{H^2(\Omega)} </math>. For the general case of <math>u\in H^2(\Omega)</math> non identically null, it suffices to apply this inequality to the function <math>u/ \|u\|_{H^1(\Omega)}</math>. }} Noticing that, for any <math>v\in H^2(\mathbb{R}^2)</math>, there holds

<math>\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,</math>

one deduces from the Brezis-Gallouet inequality that there exists <math>C>0</math> only depending on <math>\Omega</math> such that, for all <math>u\in H^2(\Omega)</math> which is not a.e. equal to 0,

<math>\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).</math>

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

See also

• Ladyzhenskaya inequality
• Agmon's inequality