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

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.