Riesz representation theorem in Sobolev spaces

Question

My question is about functionals on $W_{1,p}(\Omega)$ spaces, $\Omega$ is contained in $\mathbb R^n$

I am trying to figure out if there is a way to characterize all linear functionals on the above space.

Is there any version of Riesz representation theorem in general Banach space?

Answer 1

There is no Riesz representation theorem that would say something about a general Banach space $X$. For such a space, it is customary to denote the space of all continuous linear functionals as $X^*$, which is called the dual space of $X$. One can try to give a concrete description of the elements of $X^*$, but the success depends on what $X$ is. For many function spaces the structure of the dual is already known, as searching for "dual of ... space" will show you.

Specifically, the dual space of the Sobolev space $W^{1,p}$ is described by Theorem 3.9 of the book Sobolev spaces by Adams.

For every $L\in (W^{1,p}(\Omega))^*$ there exist elements $v_0,v_1,\dots,v_n\in L^{p'}(\Omega)$ (where $n$ is the dimension and $p'=p/(p-1)$) such that $$L(u) = \int_\Omega \left(uv_0+\sum_{k=1}^n \frac{\partial u}{\partial x_k} v_k\right)$$

The structure of the dual is nicer if one restricts the attention to $W_0^{1,p}(\Omega)$, because integration by parts does not produce boundary terms then. The dual of $W_0^{1,p}(\Omega)$ is naturally identified with $W^{-1,p'}(\Omega)$, a Sobolev space of negative order of smoothness.

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See also: Dual space of the sobolev spaces.