Work

The project is aimed at investigating the qualitative properties (in particular, problems of regularity and removability of singularities) of solutions of quasilinear elliptic and parabolic equations with non-standard growth conditions and at studying the asymptotic behavior of general metric spaces at infinity. It is planned to develop new effective methods for investigating the qualitative properties of solutions of elliptic and parabolic equations with non-standard growth conditions and to obtain conditions for regularity and removability of singularities for quasilinear equations, in particular, for anisotropic elliptic and parabolic equations with absorption term. Within the study of an asymptotic behaviour of general metric spaces, we plane to find the necessary and sufficient conditions for the geodesicity of pretangent (tangent) spaces to general metric spaces at infinity and to obtain the conditions, under which these spaces are Busemann convex and have nonpositive and nonnegative Aleksandrov curvature. The study of the qualitative properties of solutions of diffusion equations, which will be considered in the project, can serve as a theoretical basis in the study of diffusion processes of metals and alloys (in particular, semiconductors); nanotechnologies, in the modeling of electrorheological fluids, the properties of which are able to undergo significant changes in the application of the electromagnetic field, in the fields of computer vision and image processing using an anisotropic diffusion filter (smoothing, segmentation, edge detection). Investigations of the asymptotic behavior of unbounded metric spaces at infinity can afford to get a construction of models that describe the asymptotic structure of big data and the asymptotic behavior of self-learning artificial intelligence systems which learn on the basis of these data.