Agmon's inequality

In mathematical analysis, Agmon's inequalities, named after Shmuel Agmon,^[1] consist of two closely related interpolation inequalities between the Lebesgue space $L^\infty$ 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 $\Omega\subset\mathbb{R}^3$. Then Agmon's inequalities in 3D state that there exists a constant $C$ such that

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

and

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

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

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

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 $u\in H^{s_2}(\Omega)$

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

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. ISBN 978-0-8218-4910-1.

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.