The goal of the proposal is to investigate various problems in Geometric Analysis with the theory of Sobolev spaces as a main tool. As the theory of Sobolev spaces has an impressive range of applications, the proposal does not focus on a narrow problematics but rather, it addresses a number of problems from a broader range of areas in Geometric Analysis. That includes study of: (1) boundedness of maximal functions in Sobolev spaces; (2) Sobolev extension domains; (3) interplay between isoperimetric inequality, Sobolev inequality and the truncation method in the vectorial case related to Korn's inequality and the Sobolev inequality of Strauss; (4) geometric properties of Sobolev mappings between domains in the Euclidean space (this research is related to problems in nonlinear elasticity); (5) bi-Lipschitz embeddings of metric spaces; (6) Lipschitz approximation of Sobolev mappings between manifolds, polyhedra and metric spaces with connections to the topology of spaces. (7) degree theory of mappings in Orlicz-Sobolev spaces in the case in which the target space is a rational homology sphere; (8) the Hardy space regularity of Jacobians of mappings between manifolds with applications to the regularity questions in the calculus of variations.
Theory of Sobolev spaces was one of the greatest discoveries in the XXth century mathematics. This theory is the most important single tool in studying nonlinear partial differential equations, both in its theoretical aspects and numerical implementation. Although the theory of Sobolev spaces has been created in the late thirties,in recent years, there have been major breakthroughs in the theory, by expanding the applications to new areas of pure mathematics like analysis on metric spaces, geometric group theory or algebraic topology as well as to areas in applied mathematics, like for example to non-convex calculus of variations. The aim of the proposal is to continue this investigation in its various aspects. The PI has intention to advise PhD students on problems closely related to the proposal.
