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Consider a regular bounded open set $\Omega\subset\mathbb{R}^3$, and a set of regular scalar functions $(u_n)_n\in\mathscr{C}^\infty(\Omega)$ such as $\|\Delta u_n\|_{L^\infty(\Omega)} \leq C$. Is it possible to show that $\|\nabla u_n\|_{L^p_{\text{loc}}(\Omega)}$ is also bounded for some $p\in ]1,\infty[ $ ? Or the values on the boundary are too important ?

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    How is the norm in $L^p_{\operatorname{loc}}$ defined?2012-06-29
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    You're right, Davide, of course I meant $\nabla u_n$ bounded in the mentionned space (which is not normed, but is Fréchet). Anyway... this seems to be false, even in in dimension 1 (see below)2012-06-29

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