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

Abstract

The following problem is studied: describe the triplets (Ω,g,μ), μ=ρdx, where g=(gij(x)) is the (co)metric associated with the symmetric second order differential operator L(f)=1ρ∑ij∂i(gijρ∂jf) defined on a domain Ω of Rn and such that there exists an orthonormal basis of L2(μ) 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 ∂Ω contains a piece of a tangent developable surface. The proof is based on Plücker-like formulas in the form given by Ragni Piene. All the found solutions are generalized for any dimension.