Solution
= Solution
The radial <Poisson equation> with constant source one has $\psi=r^2/6+a+b/r$. The boundary conditions give
$$
\frac16+a+b=1,\qquad\frac46+a+\frac b2=1.
$$
Subtracting gives $b=1$, then $a=-1/6$. Hence
$$
\boxed{\psi(r)=\frac{r^2-1}{6}+\frac1r,\qquad1\le r\le2.}
$$
Its <radial Laplacian> is one and both endpoint values are one. The prescribed values ensure continuity with the adjacent region solutions, but do not impose derivative continuity.