Singular Sturm-Liouville Problems on a Simplex in d-Dimensions

The following section was inspired by the work of Owens [29]. In that paper he derived an orthogonal basis for approximating functions on a triangle from a basic premise. We will take an alternative route and show that it is possible to find a singular Sturm-Liouville operator for a d-dimensional simplex and present its eigenfunction/eigenvalue pairs explicitly. The eigenfunctions for the cases d=1 (a segment), d=2 (a triangle), and d=3 (a tetrahedron) will prove to be useful in the context of polynomial-based approximation on these simplices. As stated before these results have been proposed in [28].

A d-dimensional simplex S^d can be defined as a set of constraints on the entries of a d-dimensional vector:

$$S^d = \{ {\bf r} \in {\Bbb R}^d\vert 0 \leq \sum_{i=1}^{i=j} r_i \leq 1; \hspace{4pt} j=1,2 ...,d \}$$

We define an operator on a d-dimensional space of at least twice differentiable functions of d variables:

$$L^d_{\bf r} = \sum_{i=1}^{i=d} \partial_{r_i}(r_i \partial_{r_i} - r_i \sum_{j=1}^{j=d} r_j \partial_{r_j})$$

We also define a (d+1) vector, ${\bf s} = \left[(1-\omega)r_1,(1-\omega)r_2,...,(1-\omega)r_d, \omega\right]$, where $\omega \in [-1,1]$. Using the identities:

it is straightforward to show the following relationship:

$$L_{\bf s}^{d+1} = \frac{1}{1-\omega}L_{\bf r}^{d} + \frac{1}{(1-\omega)^d}\partial_\omega(\omega(1-\omega)^{d+1}).$$

From this we can repeat the recurrence relation ending up with the operator in terms of the ${\bf a}$ coordinates:

$$L_{\bf r}^{d} = \sum_{i=1}^{i=d} \frac{1}{\Pi_{j=i+1}^{j=d}(... ...l_{a_i}(a_i(1-a_i)\partial_{a_i})-(i-1)a_i\partial_{a_i}\right]$$

where ${\bf a}$ is defined by the canonical transform:

This form of the operator shows that $L_{\bf r}^{d}$ is self-adjoint in the inner product taken over the simplex S^d. This is because S^d maps to a d-dimensional unit box $U^d = \{ {\bf r} \in {\Bbb R}^d\vert 0 \leq r_i \leq 1 \hspace{4pt} i=1,2 ...,d \}$ and using integration by parts we notice that all the surface integral terms are zero.

For example, we consider the i^th operator in the sum for $L_{\bf r}^{d}$:

We can also use the definition of the Jacobi polynomials to show that $L_{\bf r}^{d}$ has eigenfunctions:

$$\phi^d_{\bf d} = \Pi_{i=1}^{i=d} (1-a_i)^{c_i} P^{2 c_i+i-1,0}_{d_i}(a_i)$$

where ${\bf d} \in {\Bbb N}^d$, $c_i = \sum_{j=0}^{i-1} d_j$. The eigenvalues for these eigenfunctions are:

$$\lambda^d_{\bf d} = c_{d+1} (c_{d+1} + d)$$

In summary, the operators we have defined in the d-dimensional simplex have simple tensor product eigenfunctions when mapped to a d-dimensional unit box. Also, their eigenvalues are n(n+d) where n is the total degree of the eigenfunction in both the mapped and original coordinates ${\bf a}$ and ${\bf r}$.