Agmon's Inequality

In mathematical analysis, Agmon's inequalities, named after Shmuel Agmon,Lemma 13.2, in: Agmon, Shmuel, ''Lectures on Elliptic Boundary Value Problems'', AMS Chelsea Publishing, Providence, RI, 2010. . consist of two closely related interpolation inequalities between the Lebesgue space $L^\infty$ and the Sobolev spaces $H^s$. It is useful in the study of partial differential equations.

Let $u\in H^2(\Omega)\cap H^1_0(\Omega)$ where $\Omega\subset\mathbb^3$. Then Agmon's inequalities in 3D state that there exists a constant $C$ such that

: $\displaystyle \, u\, _\leq C \, u\, _^ \, u\, _^,$

and

: $\displaystyle \, u\, _\leq C \, u\, _^ \, u\, _^.$

In 2D, the first inequality still holds, but not the second: let $u\in H^2(\Omega)\cap H^1_0(\Omega)$ where $\Omega\subset\mathbb^2$. Then Agmon's inequality in 2D states that there exists a constant $C$ such that

: $\displaystyle \, u\, _\leq C \, u\, _^ \, u\, _^.$

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

: $\displaystyle \, u\, _\leq C \, u\, _^ \, u\, _^$

See also

* Ladyzhenskaya inequality
* Brezis–Gallouet inequality

Notes

References

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Theorems in analysis
Inequalities

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