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.

Bibliographie

[1] A. P. Calderon, A. Zygmund, “On the existence of certain singular integrals”, Acta Math., 88:1 (1952), 85–139 | DOI | MR | Zbl

[2] N. G. Meyers, “An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations”, Ann. Scuola Norm. Sup. Pisa (3), 17 (1963), 189–206 | MR | Zbl

[3] B. V. Boyarskii, “Obobschennye resheniya sistemy differentsialnykh uravnenii pervogo poryadka ellipticheskogo tipa s razryvnymi koeffitsientami”, Matem. sb., 43(85):4 (1957), 451–503 | MR | Zbl

[4] F. W. Gehring, “The $L^p$-integrability of the partial derivatives of a quasiconformal mapping”, Acta Math., 130:1 (1973), 265–277 | DOI | MR | Zbl

[5] M. Giaquinta, G. Modica, “Regularity results for some classes of higher order non linear elliptic systems”, J. Reine Angew. Math., 311–312 (1979), 145–169 | MR | Zbl

[6] M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Ann. of Math. Stud., 105, Princeton Univ. Press, Princeton, NJ, 1983 | MR | Zbl

[7] M. Giaquinta, Introduction to regularity theory for nonlinear elliptic systems, Lectures Math. ETH Zürich, Birkhäuser, Basel, 1993 | MR | Zbl

[8] I. V. Skrypnik, Methods for analysis of nonlinear elliptic boundary value problems, Transl. Math. Monogr., 139, Amer. Math. Soc., Providence, RI, 1994 | MR | MR | Zbl | Zbl

[9] V. V. Zhikov, “On some variational problems”, Russian J. Math. Phys., 5:1 (1997), 105–116 | MR | Zbl

[10] V. V. Zhikov, “Meyers type estimates for the solution of a nonlinear Stokes system”, Differential Equations, 33:1 (1997), 108–115 | MR | Zbl

[11] V. V. Zhikov, S. E. Pastukhova, “On Gehring's lemma”, Dokl. Math., 77:2 (2008), 243–248 | DOI

[12] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, de Gruyter, Paris, 1969 | MR | MR | Zbl | Zbl

[13] V. V. Zhikov, “Solvability of the three-dimensional thermistor problem”, Proc. Steklov Inst. Math., 261:1 (2008), 98–111 | DOI

[14] V. V. Zhikov, “On Lavrent'ev's effect”, Russian Acad. Sci. Dokl. Math., 52:3 (1995), 325–329 | MR | Zbl

[15] V. V. Zhikov, “Density of smooth functions in Sobolev–Orlicz spaces”, J. Math. Sci. (N. Y.), 132:3 (2006), 285–294 | DOI | MR | Zbl

[16] E. Acerbi, G. Mingione, “Regularity results for a class of functionals with non-standard growth”, Arch. Ration. Mech. Anal., 156:2 (2001), 121–140 | DOI | MR | Zbl

[17] Yu. A. Alkhutov, “The Harnack inequality and the Holder property of solutions of nonlinear elliptic equations with a nonstandard growth condition”, Differential Equations, 33:12 (1997), 1653–1663 | MR | Zbl

[18] L. Diening, “Maximal function on Musielak–Orlicz spaces and generalized Lebesgue spaces”, Bull. Sci. Math., 129:8 (2005), 657–700 | DOI | MR | Zbl

[19] E. Acerbi, G. Mingione, “Regularity results for stationary electro-rheological fluids”, Arch. Ration. Mech. Anal., 164:3 (2002), 213–259 | DOI | MR | Zbl

[20] D. Kinderlehrer, G. Stampacchia, An introduction to variational inequalities and their applications, Pure Appl. Math., 88, Academic Press, New York–London, 1980 | MR | MR | Zbl | Zbl

[21] V. V. Zhikov, “Questions of convergence, duality, and averaging for functionals of the calculus of variations”, Math. USSR-Izv., 23:2 (1984), 243–276 | DOI | MR | Zbl | Zbl

[22] V. V. Zhikov, “On passage to the limit in nonlinear variational problems”, Russian Acad. Sci. Sb. Math., 76:2 (1993), 427–459 | DOI | MR | Zbl | Zbl

[23] V. V. Zhikov, “K tekhnike predelnogo perekhoda v nelineinykh ellipticheskikh uravneniyakh”, Funkts. analiz i ego prilozheniya, 43:2 (2009), 19–38

[24] I. Ekeland, R. Temam, Convex analysis and variational problems, Studies in Mathematics and its Applications, 1, North-Holland, Amsterdam–Oxford; Elsevier, New York, 1976 | MR | MR | Zbl | Zbl

[25] D. E. Edmunds, J. Rákosnik, “Sobolev embeddings with variable exponent”, Studia Math., 143:3 (2000), 267–293 | MR | Zbl

[26] X. Fan, J. Shen, D. Zhao, “Sobolev embedding theorems for spaces $W^{k,p(x)}(\Omega)$”, J. Math. Anal. Appl., 262:2 (2001), 749–760 | DOI | MR | Zbl

[27] W. Orlicz, “Über konjugierte Exponentenfolgen”, Stud. Math., 3 (1931), 200–211 | Zbl

[28] F.-Y. Maeda, “Poincare type inequality and an integrability result for variable exponent”, JIPAM. J. Inequal. Pure Appl. Math., 9:3 (2008), article 68