Please use this identifier to cite or link to this item: http://hdl.handle.net/10995/101668
Title: Integrability properties of functions with a given behavior of distribution functions and some applications
Свойства интегрируемости функций с заданным поведением функций распределения и некоторые приложения
Authors: Kovalevsky, A. A.
Issue Date: 2019
Publisher: Krasovskii Institute of Mathematics and Mechanics
Citation: Kovalevsky A. A. Integrability properties of functions with a given behavior of distribution functions and some applications / A. A. Kovalevsky. — DOI 10.21538/0134-4889-2019-25-1-78-92 // Trudy Instituta Matematiki i Mekhaniki UrO RAN. — 2019. — Vol. 25. — Iss. 1. — P. 78-92.
Abstract: We establish that if the distribution function of a measurable function v given on a bounded domain Ω of Rn (n > 2) satisfies, for sufficiently large k, the estimate meas{|v| > k} 6 k−αϕ(k)/ψ(k), where α > 0, ϕ: [1, +∞) → R is a nonnegative nonincreasing measurable function such that the integral of the function s → ϕ(s)/s over [1, +∞) is finite, and ψ: [0, +∞) → R is a positive continuous function with some additional properties, then |v|αψ(|v|) ∈ L1(Ω). In so doing, the function ψ can be bounded or unbounded. We give corollaries of the corresponding theorems for some specific ratios of the functions ϕ and ψ. In particular, we consider the case where the distribution function of a measurable function v satisfies, for sufficiently large k, the estimate meas{|v| > k} 6 Ck−α(ln k)−β with C, α > 0 and β > 0. In this case, we strengthen our previous result for β > 1 and, on the whole, we show how the integrability properties of the function v differ depending on which of the intervals [0, 1] or (1, +∞) contains β. We also consider the case where the distribution function of a measurable function v satisfies, for sufficiently large k, the estimate meas{|v| > k} 6 Ck−α(ln ln k)−β with C, α > 0 and β > 0. We give examples showing the accuracy of the obtained results in the corresponding scales of classes close to Lα(Ω). Finally, we give applications of these results to entropy and weak solutions of the Dirichlet problem for nonlinear elliptic second-order equations with right-hand side in some classes close to L1(Ω) and defined by the logarithmic function or its double composition. © 2019 Krasovskii Institute of Mathematics and Mechanics. All Rights Reserved.
Keywords: DIRICHLET PROBLEM
DISTRIBUTION FUNCTION
ENTROPY SOLUTION
INTEGRABILITY
NONLINEAR ELLIPTIC EQUATIONS
RIGHT-HAND SIDE IN CLASSES CLOSE TO L1
WEAK SOLUTION
URI: http://hdl.handle.net/10995/101668
Access: info:eu-repo/semantics/openAccess
SCOPUS ID: 85075216614
PURE ID: 9205789
ISSN: 1344889
DOI: 10.21538/0134-4889-2019-25-1-78-92
Appears in Collections:Научные публикации, проиндексированные в SCOPUS и WoS CC

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