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Let $f\in W_0^{1,p}(U)$, for $U$ a bounded domain and $p < n/(n-1)$. I am trying to prove that there is an inequality of the form

$$\|f\|_{W^{1,p}} \leq C \int_{\Omega} |\Delta f| $$

where $\Delta$ is the Laplacian. I tried applying the Sobolev embedding theorem, followed by the inequality

$$\|D^2 f\|_p \leq C \|\Delta f\|_p$$

but this does not hold for $p = 1$. I also tried using elliptic estimates, also to no avail. Does anyone have any suggestions?

EDIT: Perhaps it is possible to deduce this from the $L^p$ inequality

$$\|D^2 f\|_{L^p} \leq C \|\Delta f\|_{L^p}$$

for $u\in W_0^{2,p}(\Omega)$, $1 < p < \infty$?

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    Do you mean $\|f\|_{W^{1,n/(n-1)}}$?2012-11-11
  • 0
    Yes, sorry. Fixed.2012-11-11
  • 0
    I still don't see why there should exist an inequality of this form. Let $U:=(0,1)$ and $f(x) := x$, then $\Delta f = 0$, but $\|f\|_{W^{1,\frac{n}{n-1}}} >0$ ...?2012-11-11
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    Some condition is clearly needed so that the solution of $\Delta u = f$ becomes unique.2012-11-11
  • 0
    Oh, sorry, my mistake. I have corrected the question to state that $f\in W_0^{1,p}(\Omega)$ for $p < n/(n-1)$.2012-11-11

2 Answers 2