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Generalized implicit function theorems and problems with a free boundary. (English) Zbl 0860.35001
The content of the present work is as follows: In Sec. 1.1 the properties of superposition operators on Hölder classes are considered. Information on the derivatives along vector fields is presented in Sec. 1.2. Questions concerning perturbations of boundary value problems are considered in Sec. 1.3. The results of this section play a central role in the verification of assumptions of the generalized implicit function theorems. In Sec. 1.4 the parametrization of doubly-connected domains by functions defined on a definite manifold are considered.
Local and global generalized implicit function theorems are presented in Chapter 2. In Sec. 2.1. the local existence, uniqueness and differentiability of the implicit function are considered. In Sec. 2.2 global variants of the generalized implicit function theorems are presented.
In Chapter 3 and 4 classical problems are considered in view of the results of Chapter 1 and 2. Here we have conditions of global stability of these problems in Hölder classes together with data-dependence results in the dimension \(n\geq 2\). Finally, Chapter 4 is devoted to problems with Bernoulli’s conditions on a free boundary.
MSC:
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
26B10 Implicit function theorems, Jacobians, transformations with several variables
35R35 Free boundary problems for PDEs
35B20 Perturbations in context of PDEs
49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
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