84,163 research outputs found
Decomposition theorem and Riesz basis for axisymmetric potenials in the right hal-plane
The Weinstein equation with complex coefficients is the equation governing
generalized axisymmetric potentials (GASP) which can be written as
, where . We
generalize results known for to . 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 for Re . We establish a new decomposition
theorem for the GASP in any annular domains for , which is in
fact a generalization of the B\^ocher'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 , we show that this family is even a
Riesz basis in some non-concentric circular annulus
Lq-Helmholtz decomposition and Lq-spectral theory for the Maxwell operator on periodic domains
We investigate the Helmholtz decomposition on periodic domains and prove the existence of the Lq-Helmholtz decomposition on periodic domains for a suitable range of q depending on the regularity of the boundary. As applications, we prove a spectral multiplier theorem for a Maxwell-type operator and study the incompressible Navier-Stokes equations on periodic domains in the Lq-setting
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