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
![{\displaystyle \displaystyle \|u\|_{L^{\infty }(\Omega )}\leq C\|u\|_{H^{1}(\Omega )}^{1/2}\|u\|_{H^{2}(\Omega )}^{1/2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4346ae01aa46ee0e30cd99fe1c669f9fe2221172)
and
![{\displaystyle \displaystyle \|u\|_{L^{\infty }(\Omega )}\leq C\|u\|_{L^{2}(\Omega )}^{1/4}\|u\|_{H^{2}(\Omega )}^{3/4}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c68117c2909f335903fe10822d2c65f4bed55c7a)
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
![{\displaystyle \displaystyle \|u\|_{L^{\infty }(\Omega )}\leq C\|u\|_{L^{2}(\Omega )}^{1/2}\|u\|_{H^{2}(\Omega )}^{1/2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa9a6606dfa606a54d39b631e8e69a9811b91455)
For the
-dimensional case, choose
and
such that
. Then, if
and
, the following inequality holds for any
![{\displaystyle \displaystyle \|u\|_{L^{\infty }(\Omega )}\leq C\|u\|_{H^{s_{1}}(\Omega )}^{\theta }\|u\|_{H^{s_{2}}(\Omega )}^{1-\theta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a7a4c2735a000ae6ba5d77b9c3cbda7d51db9e3)
- ^ Lemma 13.2, in: Agmon, Shmuel, Lectures on Elliptic Boundary Value Problems, AMS Chelsea Publishing, Providence, RI, 2010. ISBN 978-0-8218-4910-1.