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

Slah Chaabi (1, 2) , Stephane Rigat (2)

1 APICS - Analysis and Problems of Inverse type in Control and Signal processing
2 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 Bocher's decomposition theorem. In par- ticular, 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 annulus.

Mots clés: Axially symmetric solutions; Fundamental solutions; Riesz Basis; Elliptic functions and integrals

Domaines: Variables complexes [math.CV]; Equations aux dérivées partielles [math.AP]

Slah Chaabi, Stephane Rigat. Decomposition theorem and Riesz basis for axisymmetric potentials in the right half-plane. 2014. ⟨hal-00940237v1⟩