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What is the significance of Green's function being symmetric ? How do I understand this intuitively? Thanks in advance .

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The importance of the symmetry in Green's function is that it can be used to construct Green's functions using free space Green's function as its foundations.

One example of the use of the symmetric property is in proving that Poisson's formula $u$ on the half space $\mathbb{R}^n$ belongs to the space \begin{equation*} C^{\infty}(\mathbb{R}^n_+)\cap L^{\infty}(\mathbb{R}^{n-1}) \end{equation*} (ie, smooth functions on the half plane extended to $L^{\infty}$ on the $n-1-$dimensional real numbers).

The intuition behind Green's function symmetry comes from odd/even functions. If $f:\mathbb{R}\to\mathbb{R}$ is a continuously differentiable function, then we can write two functions \begin{equation*} y(x)=f(x)-f(-x) \\ z(x)=f(x)+f(-x) \end{equation*} with initial conditions $y(0)=0$ and $z'(0)=0$. Observe that we can derive Dirichlet and Neumann type boundary conditions when we subtract mirror images of a function. This leads to the symmetry condition \begin{equation*} G(x,y)=G(y,x),~x,y\in\Omega,~x\neq y \end{equation*} where $\Omega$ is an open and bounded subset of $\mathbb{R}^n$, and $G$ is Green's function.

Does that answer your question? I'll expand or clarify anything if you like.

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