Existence of Green's function and the Dirichlet problem

Question

Let $U \subset \mathbb{R}^n$ be a bounded domain, and consider the following problem : $$\left\{ \begin{array}{lcr} -\Delta u = 0 & & \text{in } U, \\ u = g & & \text{on } \partial U, \end{array} \right.$$ where $g$ is a given function and $u \in C^2(U) \cap C(\overline{U})$.

Does the existence of Green's function for $U$ imply the existence of a solution to the above Dirichlet problem ?

I know that the existence of a solution to the above Dirichlet problem depends both on the regularity of $\partial U$ and on the choice of $g$. On the other side, Green's function is defined as $G(x,y) = \Psi(x-y) - \phi^x(y)$, $x,y \in U$ and $x \neq y$, where $\Psi$ is the fundamental solution to Laplace's equation (and thus independent of $g$) and $\phi^x$ satisfies $$\left\{ \begin{array}{lcr} \Delta \phi^x = 0 & & \text{in } U, \\ \phi^x = \Psi(y-x) & & \text{on } \partial U, \end{array} \right.$$ which is also independent of $g$. If $u \in C^2(\overline{U})$ solves the Dirichlet problem, then $$u(x) = - \int_{\partial U} g(y) \frac{\partial G}{\partial \nu}(x,y) dS(y) + \int_U f(y)G(x,y) dy, \hspace{3mm} x \in U.$$

So, I'd say no : the existence of Green's function for $U$ does not imply the existence of a solution to the above Dirichlet problem. Yet, I need someone to confirm this.

And now, the other way around...

Does the existence of a solution to the above Dirichlet problem implies the existence of Green's function for $U$ ?

Thanks in advance.

Answer 1

Your question requires some clarifications to be answered precisely. Your first statement: I know that the existence of a solution to the above Dirichlet problem depends both on the regularity of $\partial U$ and on the choice of $g$ seems to imply that you are considering a general open set $U$.

In that case, you should explain what the identity $u=g$ on $\partial \Omega$ means (in the sense of traces for regular boundaries, but in general?).

Next, you define a Green function by introducing $\phi^x$, and further down, you talk of a normal derivative of $G$, $\partial_n G$ : should we therefore understand that in fact the domain $U$ has a regular boundary, where a normal exists ?

If you are happy with a regular boundary (e.g. satisfying an exterior cone condition in dimension $\geq3$, and less in dimension $2$, see the related question here), the classical Dirichlet problem is uniquely solvable for any $g\in C^0(\bar{U})$, see Gilbarg & Trudinger Chapter 2 for example. Then, provided that the Green formula you wrote makes sense (a normal exists etc.) you have constructed a Green function as well.