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

V. V. Zhikov ; S. E. Pastukhova

Sbornik. Mathematics, Tome 199 (2008) no. 12, pp. 1751-1782

Résumé

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.