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Title: | Integrability Properties of Functions with a Given Behavior of Distribution Functions and Some Applications |

Authors: | Kovalevsky, A. A. |

Issue Date: | 2020 |

Publisher: | Pleiades Publishing Pleiades Publishing Ltd |

Citation: | Kovalevsky A. A. Integrability Properties of Functions with a Given Behavior of Distribution Functions and Some Applications / A. A. Kovalevsky // Proceedings of the Steklov Institute of Mathematics. — 2020. — Vol. 308. — P. 112-126. |

Abstract: | We establish that if the distribution function of a measurable function v defined on a bounded domain Ω in ℝn (n ≥ 2) satisfies, for sufficiently large k, the estimate meas {|v| > k} ≤ 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 either 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} ≤ 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 interval, [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} ≤ 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 second-order nonlinear elliptic equations with right-hand side in some classes close to L1(Ω) and defined by the logarithmic function or its double composition. © 2020, Pleiades Publishing, Ltd. |

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/10995/111131 |

Access: | info:eu-repo/semantics/openAccess |

SCOPUS ID: | 85085513299 |

PURE ID: | 12922370 |

ISSN: | 0081-5438 |

metadata.dc.description.sponsorship: | This work was partially supported by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University). |

Appears in Collections: | Научные публикации, проиндексированные в SCOPUS и WoS CC |

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