For the Poisson equation, we may interpret $u(x, y)$ as the displacement of a membrane, e.g., a drum skin; the inhomogeneity $f(x, y)$ in the Poisson equation represents an external forcing over the surface of the membrane.
For a specific example i.e. $\Delta u = 1$ for $x^{2} + y^{2} < 1$ and $u(x, y) = 0$ for $x^{2} + y^{2} = 1$. Then we can easily get the solution is $\displaystyle u(x,y)=\frac{x^{2}+y^{2}-1}{4}$. So how can we interpret the equation as a membrane system? I am not good at physics so I would like to learn more about the physical principles in details.