# A question on the problem of Dirichlet 2

Let $$U$$ be an open set in $$\mathbb{R}^{n}$$ with $$n\geq2$$ and $$V$$ an open set containing the boundary $$\partial U$$ of $$U$$. Suppose $$u$$ is subharmonic on $$V$$. We know that the generalized solution of Dirichlet problem $$H^{U}_{u}(x)$$ exists, i.e. is a harmonic function on $$U$$ that $$\to u(y)$$ as $$x\to y$$, for all regular point $$y\in\partial U$$. My question is: can we say that $$u(x)\leq H^{U}_{u}(x)$$ for all $$x\in V\cap U$$? Notice that the answer would be yes, if $$u$$ was subharmonic on $$U$$, and not on a neighborhood of the boundary of $$U$$ only.

The answer is no. E.g., let $$U:=\{x\in\mathbb R^n\colon|x|<1\}$$, $$V:=\{x\in\mathbb R^n\colon1/2<|x|<2\}$$ (or $$V:=\mathbb R^n\setminus\{0\}$$), $$u(x):=|x|^{2-n}-1$$ for $$x\in V$$ if $$n\ge3$$, and $$u(x):=-\ln|x|$$ for $$x\in V$$ if $$n=2$$. Then $$u$$ is harmonic and hence subharmonic on $$V$$. However, $$u>0=H_u^U$$ on $$V\cap U$$.