Sobolev Space
المؤلف:
Mazja, V.
المصدر:
Sobolev Spaces. New York: Springer-Verlag, 1985.
الجزء والصفحة:
...
4-8-2021
2133
Sobolev Space
For 
, 
an open subset of 
, ![p in [1;+infty]](https://mathworld.wolfram.com/images/equations/SobolevSpace/Inline4.gif)
and 
, the Sobolev space 
is defined by
![W^(s,p)(Omega)=<span style=]() {f in L^p(Omega): forall |alpha|<=s,partial_x^alphaf in L^p(Omega)}, " src="https://mathworld.wolfram.com/images/equations/SobolevSpace/NumberedEquation1.gif" style="height:17px; width:277px" /> |
(1)
|
where 
, 
, and the derivatives 
are taken in a weak sense.
When endowed with the norm
 |
(2)
|
is a Banach space.
In the special case 
, 
is denoted by 
. This space is a Hilbert space for the inner product
 |
(3)
|
Sobolev spaces play an important role in the theory of partial differential equations.
REFERENCES:
Mazja, V. Sobolev Spaces. New York: Springer-Verlag, 1985.
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