Agmon's inequality

In mathematical analysis, Agmon's inequalities, named after Shmuel Agmon,^[1] consist of a few closely related inequalities between the Lebesgue space $L^\infty$ and the Sobolev spaces $H^s$ . It is useful in the study of partial differential equations.

The result is stated in $\mathbb{R}^3$ only. Let $u$ be a vector-valued function, $u\in (H^2(\Omega)\cap H^1_0(\Omega))^3$ where $\Omega\subset\mathbb{R}^3$ . Then Agmon's inequalities state that there exists a constant $C$ such that

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

and

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

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.

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.

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