Brezis–Gallouët inequality

In mathematical analysis, the Brezis–Gallouët inequality,[1] 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

be the exterior or the interior of a bounded domain with regular boundary, or

Then the Brezis–Gallouët inequality states that there exists a real

only depending on

equal to 0, The regularity hypothesis on

is defined such that there exists an extension operator

such that: Let

Then, denoting by

the function obtained from

by Fourier transform, one gets the existence of

only depending on

, one writes: owing to the preceding inequalities and to the Cauchy-Schwarz inequality.

This yields The inequality is then proven, in the case

, by letting

For the general case of

non identically null, it suffices to apply this inequality to the function

Noticing that, for any

, there holds one deduces from the Brezis-Gallouet inequality that there exists

only depending on

equal to 0, The previous inequality is close to the way that the Brezis-Gallouet inequality is cited in.