Agmon's inequality

In mathematical analysis, Agmon's inequalities, named after Shmuel Agmon, consist of two closely related interpolation inequalities between the Lebesgue space [math]\displaystyle{ L^\infty }[/math] and the Sobolev spaces [math]\displaystyle{ H^s }[/math]. It is useful in the study of partial differential equations. Let [math]\displaystyle{ u\in H^2(\Omega)\cap H^1_0(\Omega) }[/math] where [math]\displaystyle{ \Omega\subset\mathbb{R}^3 }[/math]. Then Agmon's inequalities in 3D state that there exists a constant [math]\displaystyle{ C }[/math] such that

[math]\displaystyle{ \displaystyle \|u\|_{L^\infty(\Omega)}\leq C \|u\|_{H^1(\Omega)}^{1/2} \|u\|_{H^2(\Omega)}^{1/2}, }[/math]

and

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

In 2D, the first inequality still holds, but not the second: let [math]\displaystyle{ u\in H^2(\Omega)\cap H^1_0(\Omega) }[/math] where [math]\displaystyle{ \Omega\subset\mathbb{R}^2 }[/math]. Then Agmon's inequality in 2D states that there exists a constant [math]\displaystyle{ C }[/math] such that

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

For the [math]\displaystyle{ n }[/math]-dimensional case, choose [math]\displaystyle{ s_1 }[/math] and [math]\displaystyle{ s_2 }[/math] such that [math]\displaystyle{ s_1\lt \tfrac{n}{2} \lt s_2 }[/math]. Then, if [math]\displaystyle{ 0\lt \theta \lt 1 }[/math] and [math]\displaystyle{ \tfrac{n}{2} = \theta s_1 + (1-\theta)s_2 }[/math], the following inequality holds for any [math]\displaystyle{ u\in H^{s_2}(\Omega) }[/math]

[math]\displaystyle{ \displaystyle \|u\|_{L^\infty(\Omega)}\leq C \|u\|_{H^{s_1}(\Omega)}^{\theta} \|u\|_{H^{s_2}(\Omega)}^{1-\theta} }[/math]

See also

• Ladyzhenskaya inequality
• Brezis–Gallouet inequality

References

• Agmon, Shmuel (2010). Lectures on elliptic boundary value problems. Providence, RI: AMS Chelsea Publishing. ISBN 978-0-8218-4910-1. 
• Foias, Ciprian; Manley, O.; Rosa, R.; Temam, R. (2001). Navier-Stokes Equations and Turbulence. Cambridge: Cambridge University Press. ISBN 0-521-36032-3. https://archive.org/details/navierstokesequa0000unse.