  The Jacobi polynomials on the simplex are orthogonal polynomials with respect
to the weight function $W_\bg(x) = x_1^{\g_1} ... x_d^{\g_d} (1-
|x|)^{\g_{d+1}}$ when all $\g_i > -1$ and they are eigenfunctions of a second
order partial differential operator $L_\bg$. The singular cases that some, or
all, $\g_1,...,\g_{d+1}$ are -1 are studied in this paper. Firstly a complete
basis of polynomials that are eigenfunctions of $L_\bg$ in each singular case
is found. Secondly, these polynomials are shown to be orthogonal with respect
to an inner product which is explicitly determined. This inner product involves
derivatives of the functions, hence the name Sobolev orthogonal polynomials.
