Analyticity of Dirichlet-Neumann Operators on Hölder and Lipschitz Domains
Analyticity of Dirichlet-Neumann Operators on Hölder and Lipschitz Domains
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Hölder和Lipschitz域上的Dirichlet-Neumann算子分析
DOI：
10.1137/s0036141004444810
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
D. Nicholls
中科院分区：
文献类型：
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作者：
Bei Hu;D. Nicholls
本文讨论了Dirichlet-Neumann算子关于边界变形的解析性问题。在两个单独的结果中，我们证明了如果变形足够小并且属于c1 +α（任意α>）或Lipschitz函数，那么Dirichlet-Neumann算子对这种变形是解析的。这两个结果的证明都利用了Nicholls和Reitich最近为稳定的、高阶的Dirichlet-Neumann算子数值模拟所提倡的“域平坦化”变量变化。我们通过使用更专门化的函数空间扩展了它们的解析性结果，并且我们的新定理在边界正则性方面是最优的。在C1+α边界摄动的情况下，基础场也位于Holder类C1+α中，定理随后诉诸于熟悉的Schauder理论论证。相反，对于Lipschitz变形，场必须位于基于L p的Sobolev空间（w1,p），因此相关的椭圆估计来自Sobolev理论。此外，在Lipschitz域的情况下，Dirichlet-Neumann算子必须弱地重新表述，以适应这些sobolov类场所具有的边界处缺乏规则性。
In this paper we take up the question of analyticity properties of Dirichlet-Neumann operators with respect to boundary deformations. In two separate results, we show that if the deformation is sufficiently small and lies either in the class of C 1+α (any α> 0) or Lipschitz functions, then the Dirichlet-Neumann operator is analytic with respect to this deformation. The proofs of both results utilize the "domain flattening" change of variables recently advocated by Nicholls and Reitich for the stable, high-order numerical simulation of Dirichlet-Neumann operators. We extend their analyticity results through the use of more specialized function spaces, and our new theorems are optimal in terms of boundary regularity. In the case of C1+α boundary perturbations the underlying field also lies in the Holder class C 1+α and the theorem follows by appealing to familiar Schauder theory arguments. In contrast, for Lipschitz deformations the field must lie in an L p -based Sobolev space (W 1,p ), so the relevant elliptic estimates come from Sobolev theory. Additionally, in the case of Lipschitz domains, the Dirichlet-Neumann operator must be reformulated weakly in order to accommodate the lack of regularity at the boundary which these Sobolev-class fields possess.