Abstract
Using Fourier series representations of functions on axisymmetric domains, we find weighted Sobolev norms of the Fourier coefficients of a function that yield norms equivalent to the standard Sobolev norms of the function.
This characterization is universal in the sense that the equivalence constants are independent of the domain.
In particular it is uniform whether the domain contains a part of its axis of rotation or is disjoint from, but maybe arbitrarily close to, the axis.
Our characterization using step-weighted norms involving the distance to the axis is different from the one obtained earlier in the book [Bernardi, Dauge, Maday Spectral methods for axisymmetric domains, Gauthier-Villars, 1999], which involves trace conditions and is domain dependent. In this paper we also provide a complement for non cylindrical domains of the proof given in loc. cit..
Sketch of results
Let $\breve\Omega\subset\mathbb{R}^3$ be an axisymetric domain with meridian domain $\Omega$. The Fourier coefficients $u^k$ of a function $\breve u$ are defined in cylindrical coordiantes as
$$
u^k(r,z) =
\frac{1}{2\pi} \int_0^{2\pi}
\breve u(r,\theta,z) \, e^{-ik\theta}\;
d\theta,\quad k\in\mathbb{Z},\quad (r,z)\in\Omega.
$$
Let $L^{2}_{1}(\Omega)$ be the Hilbert space of functions square integrable on $\Omega$ with respect to the natural measure $2\pi r\,d r\,d z$, and let the Sobolev space $H^{m}_{1}(\Omega)$ defined as the subspace of $L^{2}_{1}(\Omega)$ of functions with finite norm
$$
\|v\|^2_{H^{m}_{1}(\Omega)} = \sum_{|\alpha|\le m}
\|\partial^{\alpha}_{(r,z)}v\|^2_{L^{2}_{1}(\Omega)} \,.
$$
Theorem
For any $m\in\mathbb{N}$ there exist positive constants $c_{m}$ and $C_{m}$ such that for any $k\in\mathbb{Z}$ and any meridian domain $\Omega\subset\mathbb{R}_{+}\times\mathbb{R}$ there holds the norm equivalence
$$
c_{m} \|\breve u\|^2_{H^m(\breve\Omega)}
\le \sum_{k\in\mathbb{Z}} \|u^{k}\|^2_{C^{m}_{(k)}(\Omega)}
\le C_{m} \|\breve u\|^2_{H^m(\breve\Omega)}
$$
where the parameter dependent norm $C^{m}_{(k)}(\Omega)$ is defined as
$$
\|v\|^2_{C^{m}_{(k)}(\Omega)} =
\sum_{\ell=0}^{\min\{|k|,m\}}
\left\|\Big(\frac{|k|}{r}\Big)^{\ell} v \right\|^2_{H^{m-\ell}_1(\Omega)}
+
\sum_{\ell=1}^{[(m-|k|)/2]}
\left\| \Big(\frac{1}{r}\partial_{r}\Big)^{\ell}
\Big(\frac{1}{r}\Big)^{|k|} v \right\|^2_{H^{m-|k|-2\ell}_1(\Omega)}
$$
The norms introduced in [Bernardi, Dauge, Maday Spectral methods for axisymmetric domains] involve $\|v \|_{H^{m}_1(\Omega)}$, and, according to parity of $m-k$, $\|\frac{1}{r} \partial^{m-1}_r v \|_{L^{2}_1(\Omega)}$. The definition of the spaces for Fourier coefficients $u^k$ given there include moreover some cancellation of traces along the rotation axis, depending on $k$. The proof in loc. cit. is performed for cylindrical $\breve\Omega$ (rectangular $\Omega$). We validate these results for polygonal $\Omega$ by an extension theorem for weighted spaces.
20 mars 2022
