Eigenfunctions of the Dirichlet Laplacian in balls I am trying to find out about the Dirichlet eigenvalues and eigenfunctions of the Laplacian on $B(0, 1) \subset \mathbb{R}^n$. 
As pointed out in this MSE post, one needs to use polar coordinates, whence the basis eigenfunctions are given as a product of solutions of Bessel functions and spherical harmonics. @Neal further points out that such considerations hold even for balls on spaces of constant curvature (see Chavel, Eigenvalues in Riemannian Geometry, Chapter 2, Section 5). 
I have one question in this matter: none of the sources say what the values of the basis eigenfunctions are at the center of the ball. Clearly, the eigenfunctions should be smooth, but then it seems that they should be zero at the center of the ball. Is that correct?

Radial eigenfunctions are not zero at the centre of the ball, while nonradial eigenfunctions are zero there.
Indeed, recall that the radial part of any basis eigenfunction is
$$
r^\frac{2-n}{2} J_{l-\frac{2-n}{2}}(\sqrt{\lambda} r),
$$
where $\lambda$ is a corresponding eignevalue.
The parameter $l \in \{0,1,\dots\}$ corresponds to spherical harmonics, i.e., $l$ is a degree of a homogeneous harmonic polynomial.
So, if $l=0$, then the eigefunction is radial, while if $l \geq 1$, then the eigenfunction is nonradial.
It is known that $J_{\nu}(x) = c x^\nu + o(x^\nu)$, where $c>0$, see, e.g., here.
Therefore, we see that if $l=0$, then
$$
r^\frac{2-n}{2} J_{-\frac{2-n}{2}}(\sqrt{\lambda} r) = c > 0 \quad \text{at } r=0,
$$
while if $l \geq 1$, then
$$
r^\frac{2-n}{2} J_{l-\frac{2-n}{2}}(\sqrt{\lambda} r) = r^\frac{2-n}{2}\left(c r^{l-\frac{2-n}{2}} + o(r^{l-\frac{2-n}{2}}) \right)  = c r^l +o(r^l) =  0 \quad \text{at } r=0.
$$
