# Agmon's inequality

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In mathematical analysis, **Agmon's inequalities**, named after Shmuel Agmon,^{[1]} consist of two closely related interpolation inequalities between the Lebesgue space and the Sobolev spaces . It is useful in the study of partial differential equations.

Let where ^{[vague]}. Then Agmon's inequalities in 3D state that there exists a constant such that

and

In 2D, the first inequality still holds, but not the second: let where . Then Agmon's inequality in 2D states that there exists a constant such that

For the -dimensional case, choose and such that . Then, if and , the following inequality holds for any

## See also[edit]

## Notes[edit]

**^**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[edit]

- 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.

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