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Sbornik: Mathematics, 2008, Volume 199, Issue 12, Pages 1751–1782
DOI: https://doi.org/10.1070/SM2008v199n12ABEH003980
(Mi sm5008)
 

This article is cited in 47 scientific papers (total in 47 papers)

Improved integrability of the gradients of solutions of elliptic equations with variable nonlinearity exponent

V. V. Zhikova, S. E. Pastukhovab

a Vladimir State Pedagogical University
b Moscow State Institute of Radio-Engineering, Electronics and Automation (Technical University)
References:
Abstract: Elliptic equations of $p(x)$-Laplacian type are investigated. There is a well-known logarithmic condition on the modulus of continuity of the nonlinearity exponent $p(x)$, which ensures that a Laplacian with variable order of nonlinearity inherits many properties of the usual $p$-Laplacian of constant order. One of these is the so-called improved integrability of the gradient of the solution. It is proved in this paper that this property holds also under a slightly more general condition on the exponent $p(x)$, although then the improvement of integrability is logarithmic rather than power-like. The method put forward is based on a new generalization of Gehring's lemma, which relies upon the reverse Hölder inequality “with increased support and exponent on the right-hand side”. A counterexample is constructed that reveals the extent to which the condition on the modulus of continuity obtained is sharp.
Bibliography: 28 titles.
Received: 26.03.2008 and 17.06.2008
Bibliographic databases:
UDC: 517.956.4
MSC: Primary 35B65; Secondary 35J60
Language: English
Original paper language: Russian
Citation: V. V. Zhikov, S. E. Pastukhova, “Improved integrability of the gradients of solutions of elliptic equations with variable nonlinearity exponent”, Sb. Math., 199:12 (2008), 1751–1782
Citation in format AMSBIB
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\by V.~V.~Zhikov, S.~E.~Pastukhova
\paper Improved integrability of the gradients of solutions of elliptic equations with variable nonlinearity exponent
\jour Sb. Math.
\yr 2008
\vol 199
\issue 12
\pages 1751--1782
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  • https://doi.org/10.1070/SM2008v199n12ABEH003980
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  • This publication is cited in the following 47 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Математический сборник Sbornik: Mathematics
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    References:79
    First page:22
     
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