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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)
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
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
Linking options:
https://www.mathnet.ru/eng/sm5008https://doi.org/10.1070/SM2008v199n12ABEH003980 https://www.mathnet.ru/eng/sm/v199/i12/p19
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Abstract page: | 937 | Russian version PDF: | 341 | English version PDF: | 21 | References: | 79 | First page: | 22 |
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