Sobolev spaces are smooth? Their dual is strictly convex?

Question

Do you know any reference which says something about the:

• Smoothness of the Sobolev space $W^{1,p}(\Omega)$ i.e. if the duality mapping $J\colon W^{1,p}(\Omega)\to W^{1,p}(\Omega)^*$ is a singleton.

• Is $W^{1,p}(\Omega)^*$ strictly convex?

Here $\Omega\subset\mathbb{R}^N$ is a bounded domain and $\infty>p>1$ is any exponent.

Answer 1

There are a bunch of great books on Banach space geometry but sadly they often do not care very much about Sobolev spaces. There is Example 2.47 in Schuster at al: Regularization Methods in Banach Spaces if you want something directly quotable, although the authors also only point to other works.

However, one can also combine a bunch of results from, say, Cioranescu: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. For example like that:

• For a reflexive Banach space $X$, $X$ is strictly convex [smooth] if and only if $X^*$ is smooth [strictly convex]. This is Corollary II.1.4. So it is enough to establish that $W^{1,p}$ is smooth.
• Theorem I.3.5 says that $X$ is smooth if and only if the norm is G-differentiable on $X \setminus \{0\}$.
• In Chapter II.4 the classical result of Clarkson is established, that the $L^p$ spaces are uniformly convex, in particular strictly convex, so they are also smooth (go back and forth between $L^p$ and its dual). In particular, their norm is G-differentiable except for in $0$.
• The norm on $W^{1,p}$ is a composition of $L^p$ norms and continuous linear operators $W^{1,p} \to L^p$ and thus inherits $G$-differentiability from the $L^p$ ones, so $W^{1,p}$ is smooth and its dual is strictly convex.