Interpolation and Monotonicity
In order to take advantage of known facts about the Dirichlet and Neumann problems, it is necessary to study transformations from one region onto another and their effects on smoothness properties of transformed functions. In later chapters, it will be shown that the solution of the Neumann problem for a spherical chip with a specified normal derivative on the flat part of the boundary can be morphed into a solution of an elliptic equation with an oblique derivative boundary condition. The method for accomplishing this is called the continuity method and requires showing that the mapping from known solutions to potential solutions is a bounded transformation. The essential problem here is to establish inequalities known as apriori inequalities which are based on the assumption that there are solutions. The passage from spherical chips to regions Ω with curved boundaries requires relating the norms of functions and their transforms. The final step involves the adaptation of the Perron-Wiener method and the extraction of convergent subsequences in certain Banach spaces called Hölder spaces. Eventually, very strong conditions will have to be imposed on Ω. So strong, in fact, that the reader might reasonably conclude that only a spherical chip, or some topological equivalent, will satisfy all the conditions. As the ultimate application will involve spherical chips, the reader might benefit from assuming that Ω is a spherical chip, at least up to the point of passing to regions with curved boundaries. But as some of the inequalities require less stringent conditions on Ω, the conditions on Ω will be spelled out at the beginning of each section. Since these conditions will not be repeated in the formal statements, it is important to refer back to the beginning of each section for a description of Ω.
KeywordsBanach Space Line Segment Open Subset Curve Boundary Cauchy Sequence
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