On Estimates of Functions in Norms of Weighted Spaces in the Neighborhoods of Singularity Points

Viktor A. Rukavishnikov () and Elena I. Rukavishnikova
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Viktor A. Rukavishnikov: Computing Center of the Far Eastern Branch of the Russian Academy of Sciences, Kim Yu Chen Str. 65, 680000 Khabarovsk, Russia
Elena I. Rukavishnikova: Computing Center of the Far Eastern Branch of the Russian Academy of Sciences, Kim Yu Chen Str. 65, 680000 Khabarovsk, Russia

Mathematics, 2025, vol. 13, issue 13, 1-10

Abstract: A biharmonic boundary value problem with a singularity is one of the mathematical models of processes in fracture mechanics. It is necessary to have estimates of the function norms in the neighborhood of the singularity point to study the existence and uniqueness of the R ν -generalized solution, its coercive and differential properties of biharmonic boundary value problems with a corner singularity. This paper establishes estimates of a function in the neighborhood of a singularity point in the norms of weighted Lebesgue spaces through its norms in weighted Sobolev spaces over the entire domain, with a minimum weight exponent. In addition, we obtain an estimate of the function norm in a boundary strip for the degeneration of a function on the entire boundary of the domain. These estimates will be useful not only for studying differential problems with singularity, but also in estimating the convergence rate of an approximate solution to an exact one in the weighted finite element method.

Keywords: weighted Sobolev spaces; singularity; estimates of function norms (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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