Sobolev generalized derivative

A locally summable generalized derivative of a locally summable function (see Generalized function).

More explicitly, if $ \Omega $ is an open set in an $ n $-dimensional space $ \mathbf R ^ {n} $ and if $ F $ and $ f $ are locally summable functions on $ \Omega $, then $ f $ is the Sobolev generalized partial derivative with respect to $ x _ {j} $ of $ F $ on $ \Omega $:

$$ \frac{\partial F }{\partial x _ {j} } ( x) = \ f ( x) ,\ x \in \Omega ,\ \ j = 1 \dots n , $$

if the following equation holds:

$$ \int\limits _ \Omega F ( x) \frac{\partial \phi }{\partial x _ {j} } \ d x = - \int\limits _ \Omega f ( x) \phi ( x) d x $$

for all infinitely-differentiable functions $ \phi $ on $ \Omega $ with compact support. The Sobolev generalized derivative is only defined almost-everywhere on $ \Omega $.

An equivalent definition is as follows: Suppose that a locally summable function $ F $ on $ \Omega $ can be modified in such a way that, on a set of $ n $-dimensional measure zero, it will be locally absolutely continuous with respect to $ x _ {j} $ for almost-all points $ ( x _ {1} \dots x _ {j-1} , x _ {j+1} \dots x _ {n} ) $, in the sense of the $ ( n - 1 ) $-dimensional measure. Then $ F $ has an ordinary partial derivative with respect to $ x _ {j} $ for almost-all $ x \in \Omega $. If the latter is locally summable, then it is called a Sobolev generalized derivative.

A third equivalent definition is as follows: Given two functions $ F $ and $ f $, suppose there is a sequence $ \{ F _ {k} \} $ of continuously-differentiable functions on $ \Omega $ such that for any domain $ \omega $ whose closure lies in $ \Omega $,

$$ \int\limits _ \omega | F _ {k} ( x) - F ( x) | dx \rightarrow 0 , $$

$$ \int\limits _ \omega \left | \frac{\partial F _ {k} ( x) }{\partial x _ {j} } - f ( x) \right | d x \rightarrow 0 ,\ k \rightarrow \infty . $$

Then $ f $ is the Sobolev generalized derivative of $ F $ on $ \Omega $.

Sobolev generalized derivatives of $ F $ on $ \Omega $ of higher orders (if they exist) are defined inductively:

$$ \frac{\partial ^ {2} F }{\partial x _ {i} \partial x _ {j} } ,\ \frac{\partial ^ {3} F }{\partial x _ {i} \partial x _ {j} \partial x _ {k} } ,\dots. $$

They do not depend on the order of differentiation; e.g.,

$$ \frac{\partial ^ {2} F }{\partial x _ {i} \partial x _ {j} } = \ \frac{\partial ^ {2} F }{\partial x _ {j} \partial x _ {i} } $$

almost-everywhere on $ \Omega $.

References

[1]	S.L. Sobolev, "Some applications of functional analysis in mathematical physics" , Amer. Math. Soc. (1963) (Translated from Russian) MR1125990 MR0986735 MR0052039 Zbl 0732.46001
[2]	S.M. Nikol'skii, "A course of mathematical analysis" , 2 , MIR (1977) (Translated from Russian) Zbl 0397.00003 Zbl 0384.00004

Comments

In the Western literature the Sobolev generalized derivative is called the weak or distributional derivative.

References

[a1]	L. Schwartz, "Théorie des distributions" , 1–2 , Hermann (1973) MR2067351 MR0209834 MR0117544 MR0107812 MR0041345 MR0035918 MR0032815 MR0031106 MR0025615 Zbl 0962.46025 Zbl 0653.46037 Zbl 0399.46028 Zbl 0149.09501 Zbl 0085.09703 Zbl 0089.09801 Zbl 0089.09601 Zbl 0078.11003 Zbl 0042.11405 Zbl 0037.07301 Zbl 0039.33201 Zbl 0030.12601
[a2]	K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5 MR0617913 Zbl 0435.46002