Agmon's inequality

In mathematical analysis, Agmon's inequalities, named after Shmuel Agmon,[1] 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{R}^3. Then Agmon's inequalities in 3D state that there exists a constant C such that

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

and

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

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{R}^2. Then Agmon's inequality in 2D states that there exists a constant C such that

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

For the n-dimensional case, choose s_1 and s_2 such that s_1< \tfrac{n}{2} < s_2. Then, if 0< \theta < 1 and \tfrac{n}{2} = \theta s_1 + (1-\theta)s_2, the following inequality holds for any u\in H^{s_2}(\Omega)

\displaystyle \|u\|_{L^\infty(\Omega)}\leq C \|u\|_{H^{s_1}(\Omega)}^{\theta} \|u\|_{H^{s_2}(\Omega)}^{1-\theta}

See also

Notes

  1. ↑ 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