Agmon's inequality

In mathematical analysis, Agmon's inequalities, named after Shmuel Agmon, consist of two closely related interpolation inequalities between the Lebesgue space $L^{\mathrm{\infty }}$ and the Sobolev spaces $H^s$. It is useful in the study of partial differential equations. Let $u\in H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })$ where $\mathrm{\Omega }\subset {\mathbb{ℝ}}^3$. Then Agmon's inequalities in 3D state that there exists a constant $C$ such that

$\Vert u{\Vert }_{L^{\mathrm{\infty }}(\mathrm{\Omega })}\le C\Vert u{\Vert }_{H^1(\mathrm{\Omega })}^{1/2}\Vert u{\Vert }_{H^2(\mathrm{\Omega })}^{1/2},$

and

$\Vert u{\Vert }_{L^{\mathrm{\infty }}(\mathrm{\Omega })}\le C\Vert u{\Vert }_{L^2(\mathrm{\Omega })}^{1/4}\Vert u{\Vert }_{H^2(\mathrm{\Omega })}^{3/4}.$

In 2D, the first inequality still holds, but not the second: let $u\in H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })$ where $\mathrm{\Omega }\subset {\mathbb{ℝ}}^2$. Then Agmon's inequality in 2D states that there exists a constant $C$ such that

$\Vert u{\Vert }_{L^{\mathrm{\infty }}(\mathrm{\Omega })}\le C\Vert u{\Vert }_{L^2(\mathrm{\Omega })}^{1/2}\Vert u{\Vert }_{H^2(\mathrm{\Omega })}^{1/2}.$

For the $n$-dimensional case, choose $s_1$ and $s_2$ such that $s_1<\frac{n}{2}<s_2$. Then, if $0<\theta <1$ and $\frac{n}{2}=\theta s_1+(1-\theta )s_2$, the following inequality holds for any $u\in H^{s_2}(\mathrm{\Omega })$

$\Vert u{\Vert }_{L^{\mathrm{\infty }}(\mathrm{\Omega })}\le C\Vert u{\Vert }_{H^{s_1}(\mathrm{\Omega })}^{\theta }\Vert u{\Vert }_{H^{s_2}(\mathrm{\Omega })}^{1-\theta }$

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.