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Information × Registration Number 0225U000989, (0122U000821) , R & D reports Title The modern problems of the geometric theory of functions and mappings popup.stage_title Відображення з оберненою нерівністю Полецького у метричних просторах, теореми про локальну і межову поведінку відображень ріманових многовидів з розгалуженням і оберненою нерівністю Полецького Head Sevostianov Yevhen O., Доктор фізико-математичних наук Registration Date 22-01-2025 Organization Zhytomyr Ivan Franko State University popup.description1 The aim of the project is to develop the theory of boundary and local behavior of mappings, as well as the development of modulus technique. We plan to further apply the theory of convergence of mappings to the Dirichlet problem with respect to the Beltrami equation. We also envisage the development of the theory of extreme problems in classes of pairwise non-intersecting domains. popup.description2  The objects of research are: mappings of the Euclidean spaces, Riemannian manifolds, factor spaces and general metric spaces, internal radii of domains, functionals, random evolutions that model physical processes; random evolutions that model economic and physical processes. The purpose of the project: development of the theory of boundary and local behavior of mappings, as well as the development of modular techniques. In the course of the implementation, further applications of the theory of convergence of mappings to the Dirichlet problem regarding the Beltrami equation are planned. The development of the theory of extremal problems on classes of pairwise disjoint domains is envisaged. Research methods: The main approach that is planned to be used in the study of mappings is the so-called geometric method, and the corresponding research tool is the modulus of families of paths (surfaces). The idea of the geometric approach is to estimate both the distorted modulus of families of paths and the modulus of families of paths in the preimage under the mapping. The novelty of the approach to conducting the research primarily lies in the application of the geometric method to mappings with the inverse Poletsky inequality, whose characteristic may turn out to be non-integrable, as well as the application of the modulus method on manifolds and general metric spaces. The novelty of the project also lies in the application of the obtained results to the problem of compactness of classes of solutions of the Dirichlet problem for the Beltrami equation. The updated geometric method in mapping theory allows to significantly enrich it with new results, and also contributes to the development of the modulus technique. The main approach when solving extremal problems on classes of domains that are pairwise disjoint is to apply the combined method of “controlling functionals” based on the piecewise-separating transformation. It is also planned to apply certain numerical methods Product Description popup.authors Ilkevych Nataliia S. Androshchuk Mariia V. Bondar Serhii A. Desiatka Viktoriia S. Dovhopiatyi Oleksandr P. Zinovchuk Andrii V. Kalenska Vitalina P. Pohorui Anatolii O. Tarhonskyi Andrii L. popup.nrat_date 2025-01-22 Close