# Difference of fundamental solutions

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!

• 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? – Ben W Oct 14 '18 at 21:29
• @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 :-) – Quoka Oct 14 '18 at 22:55
• 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? – Ben W Oct 14 '18 at 23:08
• @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. – Quoka Oct 14 '18 at 23:11