Inclusion (Boolean algebra)

In mathematical analysis, the Brezis–Gallouet inequality, 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 deivatives. 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

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534