I'm reading over Peter Lax's paper On the existence of Green's functions. He gives an argument that is roughly as follows:
- The space $B'$ of continuous functions $\phi$ for which the equation
$$\left\{\begin{align*}
\Delta u = 0 &\text{ in } \Omega\\
u(x) = \phi(x) &\text{ on } \partial\Omega
\end{align*} \right.$$
admits a classical solution is a closed subspace of $B = C^0(\partial \Omega)$.- The operators $L_x: B' \to \mathbb{R}$ given by $\phi \mapsto u(x)$ are bounded (by the Maximal principle), so Hahn-Banach lets us extend them to operators on $B$.
- Now, let $N(x)$ be the fundamental solution to the Laplace equation in the whole space. Then, $N(\cdot – x_0) \in B$ for $x_0 \in \Omega$.
- The function $G(x_0, x) = N(x – x_0) – \psi(x_0,x)$ is then the Green's function, where $\psi(x_0,x) = L_x N(\cdot – x_0)$.
I'm having trouble understanding why the function $\psi$ should be harmonic in $x$. A priori, it's not even clear that the function $x \mapsto L_xf$ should even be continuous in $x$ for arbitrary $f \in B$, much less satisfy the mean value property.
Does this argument actually show that Green's functions exist? Or do we need some other argument (like Perron's method)?
Best Answer
Yes, this paper proves the existence of Green's function (hence the title), and it doesn't depend on e.g. Perron's method.
Harmonicity of $\psi$ (in original, $k_P$) is also explained there. Here, the argument is that the operators $\Delta_{x_0}$ and $L_x$ commute when applied to $N(x-x_0)$. To see this, you should first apply the difference quotient operator $$ v(\cdot) \mapsto \frac{v(\cdot+h e_i) - v(\cdot)}{h}, $$ see that this commutes with $L_x$ and pass to the limit $h \to 0$.
Once this is done (and we know $\Delta_{x_0} \psi(x_0,x) = 0$), one only needs to check that $G(x_0,x)$ has zero boundary conditions, in other words, $$ N(x-x_0) - \psi(x_0,x) \to 0 \quad \text{when $x$ is fixed and } x_0 \to \partial \Omega. $$