Diffusion orthogonal polynomials in 3-dimensional domains bounded by developable surfaces

TL;DR

This paper characterizes 3D domains with developable boundary surfaces where certain orthogonal polynomial bases are eigenfunctions of a second-order differential operator, extending previous 2D results to three dimensions.

Contribution

It extends the classification of polynomial eigenfunction bases for differential operators to three-dimensional domains bounded by developable surfaces.

Findings

Solved the problem in 3D for usual degree with developable boundary conditions.

Derived solutions using Plücker-like formulas and Ragni Piene's framework.

Generalized solutions applicable to any dimension.

Abstract

The following problem is studied: describe the triplets $(\Omega,g,\mu)$, $\mu=\rho\,dx$, where $g= (g^{ij}(x))$ is the (co)metric associated with the symmetric second order differential operator $L(f) = \frac{1}{\rho}\sum_{ij} \partial_i (g^{ij} \rho\,\partial_j f)$ defined on a domain $\Omega$ of $\mathbb R^n$ and such that there exists an orthonormal basis of $\mathcal L^2(\mu)$ made of polynomials which are eigenvectors of $L$, and the basis is compatible with the filtration of the space of polynomials with respect to some weighted degree. In a joint paper with D. Bakry and M. Zani this problem was solved in dimension 2 for the usual degree. In the author's subsequent paper this problem was solved in dimension 2 for any weighted degree. In the present paper this problem is solved in dimension 3 for the usual degree under the condition that $\partial\Omega$ contains a piece of a tangent developable surface. The proof is based on Pl\"ucker-like formulas in the form given by Ragni Piene. All the found solutions are generalized for any dimension.