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Consider the following obstacle problem in the whole domain $\mathbb{R}^n$

min{$\Delta u$, $u$-$\phi$}=0

with prescribed boundary value $\lim_{|x|\rightarrow\infty}u(x)=0$ and $\phi$ (can be assumed to be smooth enough) is the obstalce which is strictly concave (in an open set strictly containing coincidence set)and compactly supported in $\mathbb{R}^n$ .

Can we say the coincidence set $\{u=\phi\}$ is connected? Or weakly, can we say every component of coincidence set is nonempty ? Or is there any assumption that can guarantee the coincidence set is connected?

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  • $\begingroup$ What do you mean by strictly concave? I'm having hard time believing there is any compactly supported strictly concave function, but it certainly depends on your definition. $\endgroup$ Jan 30, 2015 at 13:38
  • $\begingroup$ Thank you, I mean strictly concave in the support, or in an open set containing the coincidence set. $\endgroup$
    – user64525
    Jan 30, 2015 at 15:04

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