Weak Convergence of Composition in Sobolev Space

Question

Let $\Omega\subset \mathbb R^d$ be a bounded domain with Lipschitz boundary, $1<p<\infty$, and suppose

• $h\colon \mathbb R\to \mathbb R$ is bounded and Lipschitz,
• $u_n \rightharpoonup u$ in $W^{1,p}(\Omega)$ ($\rightharpoonup$ denoting the weak convergence).

Is it true that $h(u_n) \rightharpoonup h(u)$ in $W^{1,p}(\Omega)$?

Are there other, similar convergence principles which generalize this finding? What, if $p=1$?

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Note, that $h$ is differentiable almost everywhere and that the generalized chain rule states that for $\Omega$ as given, $h$ Lipschitz and $u\in W^{1,p}(\Omega)$ one has $h(u) = h\circ u \in W^{1,p}(\Omega)$ and $$D_i(h(u)) = h_B(u)D_iu,$$ where $h_B\colon \mathbb R\to\mathbb R$ is a Borel-measurable function such that $f_B = f'$ a.e. in $\mathbb R$.

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As an example of interest, one can take for $h$ the truncation $\tau_t\colon \mathbb R\to \mathbb R$, defined by $$\tau_t(s) = \begin{cases} t, &\text{if }s\geq t,\\ s, &\text{if }-t < s < t,\\ -t, &\text{if }s\leq -t, \end{cases}$$ which is clearly Lipschitz. Suppose for some measurable functions $u$, $u_n$ that $\tau_t(u_n) \rightharpoonup \tau_t(u)$ in $W^{1,p}(\Omega)$ for every $t>0$ and let $\varphi \in W^{1,p}(\Omega)\cap L^\infty(\Omega)$. Then, for every $\varepsilon > 0$, $$\tau_\varepsilon(u_n-\varphi)\rightharpoonup \tau_\varepsilon(u-\varphi)\quad\text{in }W^{1,p}(\Omega),$$ as, for $k=\|\varphi\|_{L^\infty(\Omega)}$, $$\tau_\varepsilon(u_n-\varphi) = \tau_\varepsilon(\tau_{\varepsilon +k}(u_n) - \varphi).$$

Answer 1

Yes, it is true that $h(u_n)\rightharpoonup h(u)$ in $W^{1,p}(\Omega)$.

As $(u_n)\rightharpoonup u$ in $W^{1,p}(\Omega)$, $(u_n)$ is bounded in $W^{1,p}(\Omega)$ and by the compact embedding $W^{1,p}(\Omega)\subset L^p(\Omega)$ we have $u_n \to u$ in $L^p(\Omega)$ und thus, up to a subseqeunce, $u_n\to u$ a.e.

This given, we infer on the one hand:

• the sequence $(h(u_n))$ is bounded in $L^{p}(\Omega)$, as $h$ is bounded and $\Omega$ is bounded,
• and $(h(u_n))$ is bounded in $W^{1,p}(\Omega)$, as $$D_i h(u_n) = h'(u_n)D_i u_n$$ is bounded in $L^p(\Omega)$, since $h'$ is bounded by the Lipschitz constant of $h$,
• thus, up to a subsequence, $h(u_n)\rightharpoonup v$ in $W^{1,p}(\Omega)$ for some $v\in W^{1,p}(\Omega)$,
• and thus, as above and up to a subsequence, $h(u_n) \to v$ a.e.

On the other hand, we have

• $h(u_n) \to h(u)$ a.e., since $h$ is continuos.

Altogheter, we obtain $v=h(u)$ and thus $h(u_n) \rightharpoonup h(u)$ in $W^{1,p}(\Omega)$. Since the limit is independent of the chosen subsequence of $(u_n)$, the weak convergence holds in fact for the original sequence.

I'm looking forward to learn about other interesting convergence principles.