How Do Equivalence Classes in Sobolev Spaces W_0^{k,p} Represent Functions?

[maze]

As I understand it, one way to define Sobolev spaces is to say they are the collection of functions with weak derivatives up to some order, and this space is called $W^{k,p}$.

On the other hand, some stuff I am reading now defines it differently. Here they define the Sobolev space $W_0^{k,p}$ as the completion of $C_0^\infty$ (the space of smooth functions with compact support) with respect to the Sobolev norm.

Now, by the definition elements of $W_0^{k,p}$ are equivalence classes of Cauchy sequences in $C_0^\infty$. How do we know that these equivalence classes actually represent functions? For example, what does it even mean to "integrate" an equivalence class?

 

[jvicens]

As a scientist familiar with Sobolev spaces, I would like to clarify the definition and properties of these spaces to address your concerns.

First, let's start with the definition of Sobolev spaces. As you mentioned, one way to define Sobolev spaces is through weak derivatives. This means that for a function f in W^{k,p}, there exists weak derivatives of order up to k that are Lebesgue integrable functions. In other words, these weak derivatives are functions that can be integrated. This is important because it allows us to define the Sobolev norm, which is used to measure the "size" of functions in these spaces.

Now, let's move on to the definition of W_0^{k,p}. This space is defined as the completion of C_0^\infty (the space of smooth functions with compact support) with respect to the Sobolev norm. This means that every element in W_0^{k,p} can be approximated by a sequence of smooth functions with compact support. The completion process ensures that we have all the "missing" functions in W_0^{k,p} that can be obtained by taking limits of these approximating sequences. Therefore, the elements in W_0^{k,p} are indeed functions.

You also asked about the meaning of "integrating" an equivalence class. In this context, integration is used to define the Sobolev norm. The Sobolev norm is defined as the sum of the L^p norms of the function and its weak derivatives up to order k. Since the weak derivatives are Lebesgue integrable functions, we can integrate them in the usual sense. This allows us to define the Sobolev norm and therefore the completion process of W_0^{k,p}.

In summary, the equivalence classes in W_0^{k,p} represent functions, and integration of these equivalence classes is well-defined through the Sobolev norm. I hope this clarifies your doubts about the definition and properties of Sobolev spaces.