In mathematical analysis, Agmon's inequalities, named after Shmuel Agmon, 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 . Then Agmon's inequalities in 3D state that there exists a constant such that
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
- ^ Lemma 13.2, in: Agmon, Shmuel, Lectures on Elliptic Boundary Value Problems, AMS Chelsea Publishing, Providence, RI, 2010. ISBN 978-0-8218-4910-1.
- 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.