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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 $p=1$?


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 $\mathbb R$.


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 $k=\|\varphi\|_{L^\infty(\Omega)}$, $$\tau_\varepsilon(u_n-\varphi) = \tau_\varepsilon(\tau_{\varepsilon +k}(u_n) - \varphi).$$

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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.

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  • $\begingroup$ Out of curiosity, could you provide a reference for the chain rule for Lipschitz functions that you use on your second bullet point? (I typically see, for instance in Evans' book, the result only stated for $C^1$ functions with bounded derivative and an attempt to extend this to general Lipschitz by approximation turns ugly real fast) $\endgroup$ – Jose27 Apr 8 '17 at 5:20
  • $\begingroup$ That seems to be a delicate matter. I have relied on "Carl, S.; Le, V. K.; Motreanu, D.: Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications; Springer Science+Business Media, New York, (2007)". The authors refer to "Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin (1983)", but there the authors merely state that the proof for the general case can be done. Another reference in the book of Carl/Le is Zeidler, E.: Nonlinear FUnctional Analysis and Its Applications, Vols. II A/B. ........... $\endgroup$ – Dreipunkt Apr 9 '17 at 9:04
  • $\begingroup$ ........Springer-Verlag, Berlin (1990)", but I don't know if there is a full proof. I have searched for some papers and I have found this one: cvgmt.sns.it/media/doc/paper/199/chainrule-july20-05.pdf They refer for simple cases on old papers. In numdam.org/article/SJL_1963-1964___3_1_0.pdf is a proof for the W_0-case, which seems to be simple. And in ams.org/journals/proc/1990-108-03/S0002-9939-1990-0969514-3/… they refer to the references [4] and [14], but I have no acces to a readable version. I'm interested in a good reference, too. $\endgroup$ – Dreipunkt Apr 9 '17 at 9:09
  • $\begingroup$ In "Absolute continuity on tracks and mappings of Sobolev spaces, M. Marcus, V. J. Mizel" which can be found, e.g., at link.springer.com/journal/205/45/4/page/1 , a proof for the chain-rule (with references to the used theorems) for a more general composition is given. $\endgroup$ – Dreipunkt Apr 11 '17 at 13:11
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    $\begingroup$ I have found the statement and the full proof at Theorem 2.1.11 in Ziemer, Weakly Differentiable Functions, 1989. $\endgroup$ – Dreipunkt May 29 '17 at 11:54

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