Variational integrals with a~wide range of anisotropy
Algebra i analiz, Tome 18 (2006) no. 5, pp. 46-71.

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Anisotropic variational integrals of $(p,q)$-growth are considered. For the scalar case, the interior $C^{1,\alpha}$-regularity of bounded local minimizers is proved under the assumption that $q\le 2p$, and a famous counterexample of Giaquinta is discussed. In the vector case, some higher integrability result for the gradient is obtained.
Keywords: anisotropic problems, regularity of minimizers.
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M. Bildhauer; M. Fuchs; X. Zhong. Variational integrals with a~wide range of anisotropy. Algebra i analiz, Tome 18 (2006) no. 5, pp. 46-71. http://geodesic.mathdoc.fr/item/AA_2006_18_5_a2/

[1] Acerbi E., Fusco N., “Partial regularity under anisotropic $(p,q)$ growth conditions”, J. Diff. Equ., 107:1 (1994), 46–67 | DOI | MR | Zbl

[2] Bildhauer M., Convex variational problems: linear, nearly linear and anisotropic growth conditions, Lecture Notes in Math., 1818, Springer, Berlin-Heidelberg-New York, 2003 | MR | Zbl

[3] Bildhauer M., Fuchs M.,, “Partial regularity for variational integrals with $(s,\mu,q)$-growth”, Calc. Var., 13 (2001), 537–560 | DOI | MR | Zbl

[4] Bildhauer M., Fuchs M., “Twodimensional anisotropic variational problems”, Calc. Var., 16 (2003), 177–186 | DOI | MR | Zbl

[5] Bildhauer M., Fuchs M., “Partial regularity for a class of anisotropic variational integrals with convex hull property”, Asymp. Anal., 32 (2002), 293–315 | MR | Zbl

[6] Bildhauer M., Fuchs M., “$C^{1,\alpha}$-solutions to non-autonomous anisotropic variational problems”, Calc. Var., 24 (2005), 309–340 | DOI | MR | Zbl

[7] Bildhauer M., Fuchs M., Mingione G., “A priori gradient bounds and local $C^{1,\alpha}$-estimates for (double) obstacle problems under nonstandard growth conditions”, Z. Anal. Anw., 20:4 (2001), 959–985 | MR | Zbl

[8] Campanato S., “Hölder continuity of the solutions of some non-linear elliptic systems”, Adv. Math., 48 (1983), 16–43 | DOI | MR

[9] Choe H. J., “Interior behaviour of minimizers for certain functionals with nonstandard growth”, Nonlinear Anal., 19 (1992), 933–945 | DOI | MR | Zbl

[10] Esposito L., Leonetti F., Mingione G., “Regularity results for minimizers of irregular integrals with $(p,q)$-growth”, Forum Math., 14 (2002), 245–272 | DOI | MR | Zbl

[11] Esposito L., Leonetti F., Mingione G., “Regularity for minimizers of functionals with $p$-$q$ growth”, Nonlinear Diff. Equ. Appl., 6 (1999), 133–148 | DOI | MR | Zbl

[12] Fusco N., Sbordone C., “Some remarks on the regularity of minima of anisotropic intergals”, Comm. Partial Differential Equations, 18 (1993), 153–167 | DOI | MR | Zbl

[13] Giaquinta M., Multiple integrals in the calculus of variations and nonlinear elliptic systems, Ann. Math. Studies, 105, Princeton University Press, Princeton, 1983 | MR | Zbl

[14] Giaquinta M., “Growth conditions and regularity, a counterexample”, Manus. Math., 59 (1987), 245–248 | DOI | MR | Zbl

[15] Giaquinta M., Modica G., “Remarks on the regularity of the minimizers of certain degenerate functionals”, Manus. Math., 57 (1986), 55–99 | DOI | MR | Zbl

[16] Hong M. C., “Some remarks on the minimizers of variational integrals with non standard growth conditions”, Boll. Un. Mat. Ital. A, 6:7 (1992), 91–101 | MR | Zbl

[17] Marcellini P., “Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions”, Arch. Rat. Mech. Anal., 105 (1989), 267–284 | DOI | MR | Zbl

[18] Marcellini P., “Regularity and existence of solutions of elliptic equations with $(p,q)$-growth conditions”, J. Diff. Equ., 90 (1991), 1–30 | DOI | MR | Zbl

[19] Morrey C. B.,, Multiple integrals in the calculus of variations, Grundlehren Math. Wiss., 130, Springer, Berlin-Heidelberg-New York, 1966 | MR | Zbl

[20] Ladyzhenskaya O. A., Ural'tseva, N. N., Linear and quasilinear elliptic equations, Nauka, M., 1964

[21] Ural'tseva N. N., Urdaletova A. B., “The boundedness of the gradients of generalized solutions of degenerate quasilinear nonuniformly elliptic equations”, Vestn. Leningr. Univ., 1983, no. 4, 50–56 | MR