agmon, inequality, mathematical, analysis, agmon, inequalities, named, after, shmuel, agmon, consist, closely, related, interpolation, inequalities, between, lebesgue, space, displaystyle, infty, sobolev, spaces, displaystyle, useful, study, partial, different. In mathematical analysis Agmon s inequalities named after Shmuel Agmon 1 consist of two closely related interpolation inequalities between the Lebesgue space L displaystyle L infty and the Sobolev spaces H s displaystyle H s It is useful in the study of partial differential equations Let u H 2 W H 0 1 W displaystyle u in H 2 Omega cap H 0 1 Omega where W R 3 displaystyle Omega subset mathbb R 3 vague Then Agmon s inequalities in 3D state that there exists a constant C displaystyle C such that u L W C u H 1 W 1 2 u H 2 W 1 2 displaystyle displaystyle u L infty Omega leq C u H 1 Omega 1 2 u H 2 Omega 1 2 and u L W C u L 2 W 1 4 u H 2 W 3 4 displaystyle 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 H 2 W H 0 1 W displaystyle u in H 2 Omega cap H 0 1 Omega where W R 2 displaystyle Omega subset mathbb R 2 Then Agmon s inequality in 2D states that there exists a constant C displaystyle C such that u L W C u L 2 W 1 2 u H 2 W 1 2 displaystyle displaystyle u L infty Omega leq C u L 2 Omega 1 2 u H 2 Omega 1 2 For the n displaystyle n dimensional case choose s 1 displaystyle s 1 and s 2 displaystyle s 2 such that s 1 lt n 2 lt s 2 displaystyle s 1 lt tfrac n 2 lt s 2 Then if 0 lt 8 lt 1 displaystyle 0 lt theta lt 1 and n 2 8 s 1 1 8 s 2 displaystyle tfrac n 2 theta s 1 1 theta s 2 the following inequality holds for any u H s 2 W displaystyle u in H s 2 Omega u L W C u H s 1 W 8 u H s 2 W 1 8 displaystyle displaystyle u L infty Omega leq C u H s 1 Omega theta u H s 2 Omega 1 theta See also EditLadyzhenskaya inequality Brezis Gallouet inequalityNotes 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 EditAgmon 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 This mathematical analysis related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Agmon 27s inequality amp oldid 931441972, wikipedia, wiki, book, books, library,