I am reading a set of notes in which the author claims that the difference between any two fundamental solutions (to the Laplacian) is harmonic. That is, if $E_1$ and $E_2$ are fundamental solutions that $E_1 - E_2$ is harmonic.

A fundamental solution for the Laplacian is a function $E \in C^2(\mathbb{R}^n\setminus\{0\})$ satisfying $$ u(0) = \int_{\mathbb{R}^n} = E(x)\Delta u(x) \,\mathrm{d}{x} $$ for all $u \in C_c^2(\mathbb{R}^n)$ (i.e. $u$ is twice continuously differentiable and compactly supported).

Apparently, we can prove that the difference of fundamental solutions is harmonic using the same argument that was used (in the notes, I will write out the argument below) to show that any fundamental solution is harmonic away from the origin.

Let $E$ be a fundamental solution to the Laplacian. Suppose for a contradiction that $\Delta E(x)\neq 0$ for some $x\in\mathbb{R}^n\setminus\{0\}$. Then there exists $\epsilon > 0$ such that $\Delta E$ is of constant sign in $B_\epsilon(x)$. We may also suppose that $0 \not\in B_\epsilon(x)$. Therefore, for any non-trivial, non-negative $u\in C^2(\mathbb{R}^n)$ which is supported in $B_\epsilon(x)$ (note that such functions indeed exist), we have $$ 0 = u(0) = \int_{B_\epsilon(x)}E(y)\Delta u(y)\,\mathrm{d}{y} = \int_{B_\epsilon(x)} u(y)\Delta E(y)\,\mathrm{d}{y} \neq 0 $$ We infer that $E$ is harmonic away from the origin.

I am struggling to rigorously prove that the difference of any two fundamental solutions to the Laplacian is indeed entire harmonic. Nay helpp would be appreciated!

  • $\begingroup$ Maybe this is a silly question, but according to your definition, a fundamental solution is undefined at the origin. So how could the difference of two fundamental solutions ever be entire? $\endgroup$ – Ben W Oct 14 '18 at 21:29
  • $\begingroup$ @BenW What I think the author means is that there exists a function $g\in C^2(\mathbb{R}^n)$ such that $g$ is harmonic and $g = E_1 - E_2$ away from the origin. Thank you for taking a look at my question :-) $\endgroup$ – Quoka Oct 14 '18 at 22:55
  • $\begingroup$ If there is an entire function $g\in C^2(\mathbb{R}^n)$ such that $g=E_1-E_2$ and $E_1-E_2$ is harmonic away from the origin, isn't there some kind of theorem that means zero is a removable singularity of $E_1-E_2$? And if so, isn't $g$ harmonic trivially? So it seems you really just want to show that zero is a removable singularity of $E_1-E_2$, right? $\endgroup$ – Ben W Oct 14 '18 at 23:08
  • $\begingroup$ @BenW Not quite. We would need to show that $E_1-E_2$ is bounded near the origin. Note that if it was sufficient to find such a function $g$, then every fundamental solution could be assumed to be entire harmonic. $\endgroup$ – Quoka Oct 14 '18 at 23:11

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