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. We also provide a complement for non cylindrical domains of the proof given in loc. cit. .
翻译:使用轴对称域函数的 Fourier 序列表示法, 我们发现Fourier 系数的加权 Sobolev 规范与函数标准 Sobolev 规范相当。 这种定性是普遍性的,因为等值常数独立于域。 特别是, 域是否包含其旋转轴的一部分, 还是与轴脱节, 也可能是任意接近轴。 我们使用与轴距离有关的分级加权规范的定性不同于先前的本书[Bernardi, Dauge, Maday, Maday “轴对轴域的特征方法”, Gauthier- Villars, 1999], 涉及痕量条件和领域依赖。 我们还对引文中提供的证据的非圆形领域提供了补充。