Decomposition theorem and Riesz basis for axisymmetric potentials in the right half-plane

Slah Chaabi (1) , Stephane Rigat (1)

1 I2M - Institut de Mathématiques de Marseille

The Weinstein equation with complex coefficients is the equation governing generalized axisymmetric potentials (GASP) which can be written as Lm[u] = ∆u + (m/x) ∂xu = 0, where m ∈ C. We generalize results known for m ∈ R to m ∈ C. We give explicit expressions of fundamental solutions for Weinstein operators and their estimates near singularities, then we prove a Green's formula for GASP in the right half-plane H + for Re m < 1. We establish a new decomposition theorem for the GASP in any annular domains for m ∈ C, which is in fact a generalization of the Bôcher 's decomposition theorem. In particular , using bipolar coordinates, we prove for annuli that a family of solutions for GASP equation in terms of associated Legendre functions of first and second kind is complete. For m ∈ C, we show that this family is even a Riesz basis in some non-concentric circular annuli.

Domaines: Variables complexes [math.CV]

Slah Chaabi, Stephane Rigat. Decomposition theorem and Riesz basis for axisymmetric potentials in the right half-plane. European Journal of Mathematics, 2015, 1 (3), pp.582-640. ⟨10.1007/s40879-015-0053-5⟩. ⟨hal-00940237v2⟩