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| | In [[mathematical analysis]], '''Agmon's inequalities''', named after [[Shmuel Agmon]],<ref>Lemma 13.2, in: Agmon, Shmuel, ''Lectures on Elliptic Boundary Value Problems'', AMS Chelsea Publishing, Providence, RI, 2010. ISBN 978-0-8218-4910-1.</ref> consist of two closely related [[interpolation inequality|interpolation inequalities]] between the [[Lp space|Lebesgue space]] <math>L^\infty</math> and the [[Sobolev space]]s <math>H^s</math>. It is useful in the study of [[partial differential equation]]s. |
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| | Let <math>u\in H^2(\Omega)\cap H^1_0(\Omega)</math> where <math>\Omega\subset\mathbb{R}^3</math>. Then Agmon's inequalities in 3D state that there exists a constant <math>C</math> such that |
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| | : <math>\displaystyle \|u\|_{L^\infty(\Omega)}\leq C \|u\|_{H^1(\Omega)}^{1/2} \|u\|_{H^2(\Omega)}^{1/2},</math> |
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| | : <math>\displaystyle \|u\|_{L^\infty(\Omega)}\leq C \|u\|_{L^2(\Omega)}^{1/4} \|u\|_{H^2(\Omega)}^{3/4}.</math> |
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| | In 2D, the first inequality still holds, but not the second: let <math>u\in H^2(\Omega)\cap H^1_0(\Omega)</math> where <math>\Omega\subset\mathbb{R}^2</math>. Then Agmon's inequality in 2D states that there exists a constant <math>C</math> such that |
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| | : <math>\displaystyle \|u\|_{L^\infty(\Omega)}\leq C \|u\|_{L^2(\Omega)}^{1/2} \|u\|_{H^2(\Omega)}^{1/2}.</math> |
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| | ==See also== |
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| | * [[Ladyzhenskaya inequality]] |
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| | * [[Brezis–Gallouet inequality]] |
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| | ==Notes== |
| | <references /> |
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| | ==References== |
| | * {{cite book | last1 = Agmon | first1 = Shmuel | title = Lectures on elliptic boundary value problems | publisher = AMS Chelsea Publishing | location = , Providence, RI | year = 2010 | isbn = 978-0-8218-4910-1 }} |
| | * {{cite book | last1 = Foias | first1 = Ciprian |last2=Manley|first2=O.|last3=Rosa|first3=R.|last4=Temam|first4=R.| title = Navier-Stokes Equations and Turbulence | publisher = Cambridge University Press | location = Cambridge | year = 2001 | isbn = 0-521-36032-3 }} |
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| | [[Category:Mathematical analysis]] |
| | [[Category:Inequalities]] |
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| | {{Mathanalysis-stub}} |
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 . Then Agmon's inequalities in 3D state that there exists a constant such that
and
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
See also
Notes
- ↑ 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
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- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
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