Sobolev Spaces different definitions

[agahlawa]

Hi,

I am studying PDEs and I am confused by the definition of Sobolev spaces as they are different in two books. I'll write the definitions and mention the points of difference which I see despite which I still can't see the difference in definitions.

1) PDEs by Lawrence Evans

Let U be an open subset of $\mathbb{R}^n$. The Sobolev space $W^{k,2}(U)$ consists of all locally summable functions $u:U \rightarrow \mathbb{R}$ such that all partial derivatives of $u$ upto order $k$ exist in $\textbf{weak sense}$ and the weak partial derivatives belong to $L^p(U)$.

2) The other definition from the book: Control of linear infinite dimensional systems by Curtain & Zwart is as follows:

For $-\infty <a<b<\infty$ we define the following subspace of $L^2(a,b)$:

$W^{k,2}(a,b):=\{u \in L^2(a,b) | u, \cdots , \frac{d ^{k-1}u}{dt^{k-1}} \text{ are absolutely continuous on } (a,b) \text{ with } \frac{d^k u}{dt^k} \in L^2(a,b). \}.$

$W^{k,2}(a,b)$ is a Sobolev space.

The obvious things to be noted:

1) In definition 1 the functions map an open subset of $\mathbb{R}^n$ to $\mathbb{R}$ where as in definition 2 the functions map an open subset of $\mathbb{R}$ to $\mathbb{R}$,
2) The definition 1 requires the functions to be locally summable but in definition 2 since the functions are in $L^2(a,b)$, they are square summable on $(a,b)$, and most importantly
3) Definition 1 talks about the derivatives in weak sense whereas definition 2 claims no such thing, hence I assume that in definition 2 the derivatives are in the classical or strong sense.

My issue is that these definitions seem very different but they must be equivalent as they are both defining Sobolev spaces. Also I cannot make sense of absolute continuity and how it comes into play.

Any help would be appreciated.

Thanks.

Adi

 

[agahlawa]

I got the answer. It's proven as an exercise.

If the Sobolev space is defined for functions $u: \mathbb{R} \rightarrow \mathbb{R}$ instead of $u : \mathbb{R}^n \rightarrow \mathbb{R}$, then the conditions can be reduced to the absolute continuity of the functions.

Adi