Mgr. Eva Pernecká, Ph.D.

2018 - 2021

Non-linear functional analysis deals with problems concerning mappings of various regularity between Banach spaces and their subsets. In particular, it studies the characterization of Banach spaces by their metric structure and the stability of certain properties under non-linear homeomorphisms or embeddings. One of very important tools is so-called Lipschitz-free Banach spaces, the preduals of spaces of Lipschitz functions, which provide an abstract linearization of Lipschitz maps between metric spaces. In our research, we plan to study some open questions concerning the structure (universality, embeddings, weak sequential completeness, complementability) and approximation properties of Lipschitz-free spaces. Next, we plan to address the question whether a Sobolev homeomorphism can be approximated by diffeomorphisms or piecewise affine homeomorphisms, which is a problem of great significance in non-linear elasticity theory, PDE's and calculus of variations.

2022 - 2024

Free Banach spaces play an important role in the study of nonlinear geometry of Banach spaces and in areas involving transportation problems. This brings them to the forefront of modern functional analysis and interest in them continues to grow. The free space of a metric space M is given by the fact that M isometrically embeds in it and that any Lipschitz map from M to a Banach space extends uniquely to a bounded linear map from the free space. This allows the linearization of nonlinear problems, but at the cost of a complicated linear structure. Its closer examination is the aim of our project. The free space lies in the dual of the space of Lipschitz functions and forms its predual. Hence, one should analyze all three of the spaces to understand free space. We intend to study representation of functionals on the spaces of Lipschitz functions and topological aspects of isomorphic free spaces. This could contribute to solving some known open problems in the area, such as the complementability in the bidual, predual uniqueness, or the existence of isomorphisms of certain free spaces.