Brezis–Gallouet inequality

In mathematical analysis, the Brezis–Gallouet inequality,[1] named after Haïm Brezis and Thierry Gallouet, 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 \(u\in H^2(\Omega) \) where\( \Omega\subset\mathbb{R}^2 \). Then the Brézis–Gallouet inequality states that there exists a constant C such that

\( \displaystyle \|u\|_{L^\infty(\Omega)}\leq C \|u\|_{H^1(\Omega)}\left(1+\log\frac{\|\Delta u\|}{\lambda_1\|u\|_{H^1(\Omega)}}\right)^{1/2}, \)

where \( \Delta \) is the Laplacian, and \( \lambda_1 \) is its first eigenvalue.
See also

Ladyzhenskaya inequality
Agmon's inequality

Notes

Nonlinear Schrödinger evolution equation, Nonlinear Analysis TMA 4, 677. (1980)

References

Foias, Ciprian; Manley, O.; Rosa, R.; Temam, R. (2001). Navier–Stokes Equations and Turbulence. Cambridge: Cambridge University Press. ISBN 0-521-36032-3.