On the existence of Green's functions by Peter Lax

Question

I'm reading Peter Lax's paper On the existence of Green's functions. He showed that the regular part of Green's functions is continuous on the boundary. My question is : to have the normal derivative of Green's functions on the boundary, shouldn't we further prove the differentiability of the regular part on the boundary?

Answer 1

There's a difference between the Green's function $G(x, y)$ which solves $$ \begin{cases} -\Delta G(x, \cdot) = \delta_x & \text{ on } \Omega \\ G(x, \cdot) = 0 & \text{ on } \partial \Omega \end{cases} $$ and the Poisson kernel $K(x, y)$, which solves $$ \begin{cases} -\Delta K(x, \cdot) = 0 & \text{ on } \Omega \\ K(x, \cdot) = \delta_x & \text{ on } \partial \Omega. \end{cases} $$ I state these with delta functions informally, just to highlight the distinction.

The Green's function exists on basically any domain, but might fail to be continuous up to the boundary on a "small" set of points (see: Wiener criterion). It can be used to construct solutions to the Poisson equation $$ \begin{cases} -\Delta u = f & \text{ on } \Omega \\ u = 0 & \text{ on } \partial \Omega \end{cases} $$ by taking $u(y) = \int_\Omega G(x, y) f(x) dy$.

The Poisson kernel always exists in an abstract sense, but whether it is actually representable by a function $K(x, y)$ depends on the domain's smoothness in nontrivial ways. This is studied in the literature under the term harmonic measure. The Poisson kernel can be used to construct solutions to the Dirichlet problem $$ \begin{cases} -\Delta u = 0 & \text{ on } \Omega \\ u = g & \text{ on } \partial \Omega \end{cases} $$ using $u(y) = \int_{\partial \Omega} K(x, y) g(x) dx$.

There is a formal relationship between $G$ and $K$: $\partial_{\nu} G(x, y) = K(x, y)$ where $\nu$ is an inward unit normal and the derivative is taken in the $x$ variable. If the domain is quite smooth, the Green's function will be differentiable and the relationship takes on a pointwise meaning. If not, $G$ and $K$ may still exist (possibly in some weaker sense), and may still be related via weaker forms of this relation; this is a topic studied in harmonic analysis.

So: no, you can talk about the existence of the Green's function without any reference to its normal derivative. You can also talk about the existence of the Poisson kernel without direct use of the Green's function or its normal derivative.

If you want a more concrete answer about what you are reading, I suggest asking more specific questions.