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 is : Do we have $U\in H^2(\Omega)?$.
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Note that in your assumption $u$ itself need not be in $H^2(\Omega)$, nor even $L^2(\Omega)$, even if it is harmonic (so $u=U$). You are possibly interested on the local behaviour, that is whether $U\in H^2_{loc}$. But note that for $n=1$ any smooth, convex function is subharmonic; the function $U$ is affine on $(a-r,a+r)$, and in general not in $C^1$ (unless $u$ was already harmonic ), hence not $H^2_{loc}$. However, I think in your assumptions it is true that $U\in H^1_{loc}$. |
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Subharmonicity of this modification is true and easy to prove just from the definition of a subharmonic function. Condition $C^2$ is redundant. Of course the modification will not be in $C^2$. Now what is $H(\Omega)$ ? |
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