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Let $u\in C^2(\Omega)$ be such that $\Delta u \ge 0$ on $\Omega\supset \overline{B(a,r)}$. We consider the Poisson modification $U$ of $u$ for the ball $B(a,r),$ that is $U$ equals $u$ on $\Omega-B(a,r)$ and that on $B=B(a,r)$ equals the solution to Direchlet problem with boundary data $u|_{\partial B}$, which is given by the Poisson kernel classically denoted by $P(x,y)$. It is known that $U$ is subharmonic in the sense that it verifies an inequality mean property. My question : Do we have $U\in H^2(\Omega)?$.

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    My guess is no in general: Consider the one-dimensional case with $\Omega = \mathbb{R}$ (you can pick a bounded interval if you want) and $u(x)=x^2$, take $B(0,1)$ then $U(x)=x^2$ if $|x|\geq1$ and $U(x)=1$ if $|x|<1$. Then $U\notin H^2 (\Omega)$ (or any interval containing $[-1,1]$).2012-08-13
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    @Jose27 You are right, and the same example works in higher dimensions. Please post as an answer. To med: $H^1$ is preserved, but you probably already knew that.2012-08-13
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    The question was (unnecessarily) cross-posted on MO: http://mathoverflow.net/questions/104593/poisson-modification-of-subharmonic-function2012-08-13

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