Green's representation for the Helmholtz equation in arbitrary dimension

Question

I am looking for a reference containing a proof of Green's representation formula in $\mathbb{R}^n$ $$u(x) = \int_{\partial D} \frac{\partial u}{\partial \nu} \Phi_x - u\frac{\partial \Phi_x}{\partial \nu} \, dS,$$ where $u$ is a solution of the $n$-dimensional Helmholtz equation, $\Phi_x$ the radiating fundamental solution centred at $x$ and $D$ a suitable domain with unit normal $\nu$.

I have only been able to find a proof for $n=3$: Theorems 2.1 and 2.5 of this book.

Answer 1

The appearance of green function and theorem for PDEs has a standard approach. You can find analytical proofs in mathematical textbooks like L.Evans book (chapter 2), but specific proofs especially with vector calculus analysis are more likely to be found in physics literatures. for examplt see :

1. Green’s Functions in Classical Physics by Rother
2. Scalar wave theory, Green's function and applications by DeSanto
3. Classical Electrodynamics by jackson

proof for Helmholtz is not different from general Sturm-Liouvill problem or the Laplace PDE, that's why no book will specifically mention it.

---

But since there's standard pattern for proof of this, I'm going to write it here;

Assume $\phi_{x'}$ satisfies the Helmholtz equation, $$ \tag{1} \label{1} \Delta \phi _ {x'}(x) + k^2 \phi _ {x'}(x) = \delta(x-x') \quad x \in \mathcal{D} \subset\mathbb{R}^n, $$ where $\delta(x-x')$ is dirac delta function, i.e. $$ \tag{2}\label{2} \int_\mathcal{D'} f(x)\delta (x-x') d \mathbf{x} = f(x') $$ for all (continuous compactly supported) functions $f$, where $x' \in \mathcal{D'}$.

from green's identities, we have $$ \tag{3} \label{3} \begin{align} \int_{\mathcal{D}} \big( \psi \Delta \phi - \phi \Delta \psi \big) d\mathbf{x} &= \int_{\partial \mathcal{D}} (\phi \nabla\psi - \psi \nabla\phi ).d\mathbf{S} \\ & = \int_{\partial \mathcal{D}} (\phi \frac{\partial \psi}{\partial \nu} - \psi \frac{\partial \phi}{\partial \nu})ds. \end{align} $$ you can find proof of this in calculus or vector analysis books.
Consider $u$ is solution of homogenous Helmholtz equation, $$ \tag{4}\label{4} \Delta u + k^2u=0. $$ Start with $\ref{3}$ for $u$ and $\phi_{x'}$ and use $\ref{1}$ and $\ref{4}$, we have:

$$ \begin{align} \int_{\mathcal{D}} (u \Delta \phi_{x'} - \phi_{x'} \Delta u ) d \mathbf{x} &= \int_{\mathcal{D}} \bigg[ u . \underbrace{(-k^2 \phi_{x'} + \delta(x-x'))}_{\Delta \phi_{x'}} - \phi_{x'}. \underbrace{(-k^2 u)}_{\Delta u} \bigg] d \mathbf{x} \\ & = \int_{\mathcal{D}} u(x) \delta(x-x') d \mathbf{x} \\ \stackrel{\text{using }\ref{2}}{=} u(x') &= \int_{\partial \mathcal{D}} (\phi_{x'} \frac{\partial u}{\partial \nu} - u \frac{\partial \phi_{x'}}{\partial \nu} )ds. \end{align} $$