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I have a problem in understanding the definition of Green's function which occurs when solving a Poisson equation $\Delta u=f$. Here is the definition of our lecture:

Let $\Omega\subset\mathbb{R}^n$ be bounded with regulary domain (which means that $\partial\Omega$ may be parametrized a sufficiently smooth). A function $G:\overline{\Omega}\times\Omega\rightarrow\mathbb{R}$ is called Green's function if $$\Delta_y G(x,y)=\delta(x-y),\ \ \ x\in\Omega\\ G(x,y)=0,\ \ \ x\in\partial\Omega. $$ Here $\delta$ denotes the Dirac-delta distribution with pole in $0$, i.e. $\delta[\varphi]=\varphi(0)$ for all $\varphi\in\mathcal{C}^\infty_c(\Omega)$.

I do not understand how to interpret the first of these equations: The left-hand side is an ordinary function while the right-hand side is a distribution, presumably $\delta_x$. This does not make sense to me at all! Can someone explain this to me? Thanks a lot!

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$G$ itself is a function, but the derivative $\Delta_y$ is taken in the sense of distributions, so the resulting object $\Delta_y G(x,y)$ need not be a function.

(A simpler example: the Heaviside function $H(x)$ is a function, and its derivative in the ordinary sense is zero for all $x\neq 0$ and undefined at the origin. But one can instead interpret the derivative weakly, and find that $H$ has the Dirac delta as its distributional derivative: $H'=\delta$.)

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