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Away from the origin the gradients are $\nabla u=\hat{\mathbf{r}}/a$ and $\nabla v=-\hat{\mathbf{r}}/r^2$. The improper volume integral is thereforeThe function $v=1/r$ is a harmonic function only off the origin. It is singular inside the integration volume, so it does not meet part (a)'s regularity hypothesis. On the punctured ball, Green's first identity has outer flux $-4\pi$ and inner flux $4\pi \epsilon /a$: the inner outward unit normal is $-\hat{\mathbf{r}}$, so $u{\partial }_{n}v=(\epsilon /a)/{\epsilon }^2$. Their sum is $-4\pi +4\pi \epsilon /a$, exactly the integral before taking the limit. Equivalently the distributional Laplacian of $1/r$ is $-4\pi {\delta }_0$, rather than zero throughout the ball. This is boundary flux from a singular harmonic potential; it explains why applying part (a) across the origin would be invalid.