Boundary Layers on Sobolev–Besov Spaces and Poisson's Equation for the Laplacian in Lipschitz Domains
Boundary Layers on Sobolev–Besov Spaces and Poisson's Equation for the Laplacian in Lipschitz Domains
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DOI：
10.1006/jfan.1998.3316
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发表时间：
1998-11
影响因子：
1.7
通讯作者：
E. Fabes;O. Méndez;M. Mitrea
中科院分区：
文献类型：
--
作者：
E. Fabes;O. Méndez;M. Mitrea
本文研究了Sobolev-Besov空间中任意Lipschitz域上Laplacian算子的非齐次边值问题。因此，这是[Jerison和Kenig，J. Funct. Anal.（1995），16-219]，其中通过调和测度技术处理非齐次Dirichlet问题。我们的方法的新奇在于系统地使用边界积分方法。在这方面，关键的结果是建立可逆性的经典层潜在的运营商规模的Sobolev-Besov空间Lipschitz边界的最佳范围内的指标。本文还讨论了Lipschitz域上向量场的基于Lp的Helmholtz型分解的应用。
We study inhomogeneous boundary value problems for the Laplacian in arbitrary Lipschitz domains with data in Sobolev–Besov spaces. As such, this is a natural continuation of work in [Jerison and Kenig,J. Funct. Anal.(1995), 16–219] where the inhomogeneous Dirichlet problem is treated via harmonic measure techniques. The novelty of our approach resides in the systematic use of boundary integral methods. In this regard, the key results are establishing the invertibility of the classical layer potential operators on scales of Sobolev–Besov spaces on Lipschitz boundaries for optimal ranges of indices. Applications toLp-based Helmholtz type decompositions of vector fields in Lipschitz domains are also presented.