Registration Number
0223U000348,
0121U111851
, R & D reports
Title
Regularity for solutions of elliptic and parabolic equations and asymptotic properties of metric spaces at infinity
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Head
Bilet Viktoriia V., Кандидат фізико-математичних наук
Registration Date
06-01-2023
Organization
Institute of Applied Mathematics and Mechanics National Academy of Science of Ukraine
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The project focuses on the development of methods for studying the qualitative properties of solutions of quasilinear elliptic and parabolic equations with nonstandard growth conditions and studying the asymptotic behavior of general metric spaces at infinity. We describe the class of graphs for which all metric spaces with diametrical graphs belonging to this class are ultrametric. It is shown that a metric space is ultrametric iff the diametrical graph of the metric is either complete multipartite or empty. A refinement of the last result is obtained for totally bounded spaces. Moreover, using complete multipartite graphs we characterize the compact ultrametrizable topological spaces. We also investigate the conditions under which pseudometric spaces are complete. In particular, it was generalized and expanded some well-known characterizations of complete metric spaces to pseudometric ones. We introduced the conditions under which pseudometric spaces are pseudoisometric. It was proved that a pseudometric space is complete if and only if this space is pseudoisometric to a complete pseudometric space. We study nonnegative weak solutions of a quasilinear parabolic equations in a divergent form one of the model cases of which is anisotropic evolution p-Laplacian equation. New precise integral estimates, pointwise estimates near an isolated singularity and condition of the removability of isolated singularity were obtained. The equation with generalized Orlich growth and non-logarithmic conditions on the coefficients, which includes new cases of equations with (p,q) nonlinearity, is studied. The asymptotic behavior of the solutions was investigated and an analogue of the cluster lemma was obtained, and the Harnack-type inequality was also established.
Product Description
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Savchenko Mariia O.
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2023-01-06
