Full asymptotics and Laurent series of layer potentials for Laplace's equation on the half-space
Full asymptotics and Laurent series of layer potentials for Laplace's equation on the half-space
复制标题
半空间上拉普拉斯方程的完全渐近和层势的洛朗级数
DOI：
10.1002/mana.201800145
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发表时间：
2019
影响因子：
1
通讯作者：
Fritzsch K
中科院分区：
文献类型：
--
作者：
Fritzsch K
我们将正态分布的演算，特别是后拉定理和前推定理，与层势的方法结合起来，解决了半空间上的Dirichlet和Neumann问题。我们得到了解的完全渐近展开式，证明了边界层势算子是完全微积分的元素，并给出了经典跳跃关系的一个新的证明。在此过程中，我们改进了Siegel和Talvila对多均匀边界数据情况下修正层势的增长估计。我们在这里使用的技术可以推广到几何上更复杂的设置，例如触摸域的外部域或具有纤维尖的域。这项工作旨在成为一个更长的计划的第一步，该计划旨在理解某些非Lipschitz奇点设置中的层势方法，这些奇点可以用带有角的流形在Melrose意义上解决，并将匹配渐近分析应用于相关问题的奇异摄动。
We combine the calculus of conormal distributions, in particular the Pull‐Back and Push‐Forward Theorems, with the method of layer potentials to solve the Dirichlet and Neumann problems on half‐spaces. We obtain full asymptotic expansions for the solutions, show that boundary layer potential operators are elements of the fullb‐calculus and give a new proof of the classical jump relations. En route, we improve Siegel and Talvila's growth estimates for the modified layer potentials in the case of polyhomogeneous boundary data. The techniques we use here can be generalised to geometrically more complex settings, as for instance the exterior domain of touching domains or domains with fibred cusps. This work is intended to be a first step in a longer program aiming at understanding the method of layer potentials in the setting of certain non‐Lipschitz singularities that can be resolved in the sense of Melrose using manifolds with corners and at applying a matching asymptotics ansatz to singular perturbations of related problems.