Agmon's inequality explained

L ^{infty} and the Sobolev spaces H ^{s} . It is useful in the study of partial differential equations.

Let

u\in H ^{2(\Omega)\cap} H ^{1} _{0 (\Omega)} where \Omega\subset R ^{3} . Then Agmon's inequalities in 3D state that there exists a constant C such that\displaystyle \|u\| _{L ^{infty(\Omega)}} \leq C \|u\| ^{1 / 2} _{H ^{1(\Omega)}} \|u\| ^{1 / 2} _{H ^{2(\Omega)}} ,

and

\displaystyle \|u\| _{L ^{infty(\Omega)}} \leq C \|u\| ^{1 / 4} _{L ^{2(\Omega)}} \|u\| ^{3 / 4} _{H ^{2(\Omega)}} .

In 2D, the first inequality still holds, but not the second: let

u\in H ^{2(\Omega)\cap} H ^{1} _{0 (\Omega)} where \Omega\subset R ^{2} . Then Agmon's inequality in 2D states that there exists a constant C such that\displaystyle \|u\| _{L ^{infty(\Omega)}} \leq C \|u\| ^{1 / 2} _{L ^{2(\Omega)}} \|u\| ^{1 / 2} _{H ^{2(\Omega)}} .

For the

n -dimensional case, choose s _{1} and s _{2} such that s _{1<} \tfrac{n}{ 2 } < s _{2} . Then, if 0 < \theta < 1 and \tfrac{n}{ 2 } = \theta s _{1} + ( 1- \theta)s _{2} , the following inequality holds for any u\in H ^{s _{2}} (\Omega) \displaystyle \|u\| _{L ^{infty(\Omega)}} \leq C \|u\| ^{\theta} _{H ^{s _{1}} (\Omega)} \|u\| ^{1- \theta} _{H ^{s _{2}} (\Omega)}

See also

• Ladyzhenskaya inequality
• Brezis–Gallouet inequality

Notes

1. Lemma 13.2, in: Agmon, Shmuel, Lectures on Elliptic Boundary Value Problems, AMS Chelsea Publishing, Providence, RI, 2010. .

References

• Book: Agmon . Shmuel . Lectures on elliptic boundary value problems . AMS Chelsea Publishing . Providence, RI . 2010 . 978-0-8218-4910-1 .
• Book: Foias . Ciprian . Manley . O. . Rosa . R. . Temam . R. . Navier-Stokes Equations and Turbulence . Cambridge University Press . Cambridge . 2001 . 0-521-36032-3 . registration .