Part 1. I know how to find and prove the Green's function for Laplace operator with Dirichlet boundary condition. However, I have a question in this regard. It is well known that if $\Omega$ is smooth, bounded domain, then for any function $u \in C^2(\bar \Omega)$, we have $$u(\xi) = \int_{\partial \Omega} \left(u(x) \frac{\partial G(x ,\xi)}{\partial n_x} \right) dS_x$$ where $G(x,\xi)$ is the Green function, and $n_x$ is the unit exterior normal vector, and the formula is known as Poisson integral. The smooth and bounded requirement on $\Omega$ is coming from Green's identities, which itself rooted in divergence theorem. My question here is that in MCOWEN's book page 119, the above formula is used on $\Omega = \mathbb{R}^n_{+} = \{x \in \mathbb{R}^n: x_n >0 \}$, i.e. the upper half space, to calculate the solution to the Laplace problem with Dirichlet boundary. In fact, $n_x = - x_n$ and the calculation is straightforward. This is while such $\Omega$ set is not bounded. How do they manage to use the formula without boundedness?

Part 2. In the construction of Green's function, we mentioned that if we know that the problem \begin{equation*} \begin{aligned} \Delta u &= 0 , \quad \text{in } \Omega,\\ u & = g, \quad \text{on } \partial \Omega \end{aligned}\end{equation*} has a solution $u \in C^2(\bar \Omega)$, then this solution should satisfies \begin{equation*} u(\xi) = \int_{ \partial \Omega} g(x) \frac{\partial G(x, \xi)}{\partial n_x} dS_x. \end{equation*} However, when we find the solution for Dirichlet problem using Green's function, both in half space and on a ball, the solution $u \in C^{\infty }(\Omega) \cap C(\bar \Omega)$. This means we don't get $C^2(\bar \Omega)$, and not even $C^2(\Omega) \cap C^1(\bar \Omega)$. Is this not contradiction to the constructino of the problem?

Thanks in advance for your help.


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