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On Elliptic Homogeneous Differential Operators in Grand Spaces

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Abstract

We give an application of so-called grand Lebesgue and grand Sobolev spaces, intensively studied during last decades, to partial differential equations. In the case of unbounded domains such spaces are defined using so-called grandizers. Under some natural assumptions on the choice of grandizers, we prove the existence, in some grand Sobolev space, of a solution to the equation Pm(D)u(x) = f(x), x ∈ ℝn, m < n, with the right-hand side in the corresponding grand Lebesgue space, where Pm(D) is an arbitrary elliptic homogeneous in the general case we improve some known facts for the fundamental solution of the operator Pm(D): we construct it in the closed form either in terms of spherical hypersingular integrals or in terms of some averages along plane sections of the unit sphere.

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Funding

The work was supported by Russian Foundation for Basic Research in the framework of scientific project no. 18-01-00094-A.

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Authors and Affiliations

  1. Academy Sciences of the Chechen Republic, Kh. Ibrahimov Complex Scientific Research Institute of the Russian Academy of Sciences, 13 M. Esembaev Ave., Grozny, 364024, Russia

    S. M. Umarkhadzhiev

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  1. S. M. Umarkhadzhiev
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Correspondence to S. M. Umarkhadzhiev.

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Russian Text © The Author(s), 2020, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2020, No. 3, pp. 64–73.

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Umarkhadzhiev, S.M. On Elliptic Homogeneous Differential Operators in Grand Spaces. Russ Math. 64, 57–65 (2020). https://doi.org/10.3103/S1066369X20030056

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  • DOI: https://doi.org/10.3103/S1066369X20030056

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