A regularity theory for scalar local minimizers of splitting-type variational integrals
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 6 (2007) no. 3, p. 385-404

Starting from Giaquinta’s counterexample [12] we introduce the class of splitting functionals being of $\left(p,q\right)$-growth with exponents $p\le q<\infty$ and show for the scalar case that locally bounded local minimizers are of class ${C}^{1,\mu }$. Note that to our knowledge the only ${C}^{1,\mu }$-results without imposing a relation between $p$ and $q$ concern the case of two independent variables as it is outlined in Marcellini’s paper [15], Theorem A, and later on in the work of Fusco and Sbordone [10], Theorem 4.2.

Classification:  49N60
@article{ASNSP_2007_5_6_3_385_0,
author = {Bildhauer, Michael and Fuchs, Martin and Zhong, Xiao},
title = {A regularity theory for scalar local minimizers of splitting-type variational integrals},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 6},
number = {3},
year = {2007},
pages = {385-404},
zbl = {1150.49015},
mrnumber = {2370266},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2007_5_6_3_385_0}
}

Bildhauer, Michael; Fuchs, Martin; Zhong, Xiao. A regularity theory for scalar local minimizers of splitting-type variational integrals. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 6 (2007) no. 3, pp. 385-404. http://www.numdam.org/item/ASNSP_2007_5_6_3_385_0/

[1] R. A. Adams, “Sobolev Spaces", Academic Press, New York-San Francisco-London, 1975. | MR 450957 | Zbl 1098.46001

[2] M. Bildhauer, “Convex Variational Problems: Linear, Nearly Linear and Anisotropic Growth Conditions", Lecture Notes in Mathematics 1818, Springer, Berlin-Heidelberg-New York, 2003. | MR 1998189 | Zbl 1033.49001

[3] M. Bildhauer and M. Fuchs, Elliptic variational problems with nonstandard growth, International Mathematical Series 1, In: “Nonlinear problems in mathematical physics and related topics I, in honor of Prof. O.A. Ladyzhenskaya”, T. Rozhkovskaya (ed.), Novosibirsk, Russia, March 2002 (in Russian), 49-62; Kluwer/Plenum Publishers, June 2002 (in English), 53-66. | MR 1970604 | Zbl 1054.49026

[4] M. Bildhauer and M. Fuchs, Higher integrability of the gradient for vectorial minimizers of decomposable variational integrals, Manuscripta Math. 123 (2007), 269-283. | MR 2314085 | Zbl 1120.49031

[5] M. Bildhauer, M. Fuchs and G. Mingione, A priori gradient bounds and local ${C}^{1,\alpha }$-estimates for (double) obstacle problems under nonstandard growth conditions, Z. Anal. Anwendungen 20 (2001), 959-985. | MR 1884515 | Zbl 1011.49024

[6] M. Bildhauer, M. Fuchs and X. Zhong, Variational integrals with a wide range of anisotropy, Algebra i Analiz. 18 (2006), 46-71. | MR 2301040 | Zbl pre05232563

[7] H. J. Choe, Interior behaviour of minimizers for certain functionals with nonstandard growth, Nonlinear Anal. 19.10 (1992), 933-945. | MR 1192273 | Zbl 0786.35040

[8] L. Esposito, F. Leonetti and G. Mingione, Regularity for minimizers of functionals with $p$-$q$ growth, Nonlinear Differential Equations Appl. 6 (1999), 133-148. | MR 1694803 | Zbl 0928.35044

[9] L. Esposito, F. Leonetti and G. Mingione, Regularity results for minimizers of irregular integrals with $\left(p,q\right)$-growth, Forum. Math. 14 (2002), 245-272. | MR 1880913 | Zbl 0999.49022

[10] N. Fusco and C. Sbordone, Some remarks on the regularity of minima of anisotropic integrals, Comm. Partial Differential Equatins 18 (1993), 153-167. | MR 1211728 | Zbl 0795.49025

[11] M. Giaquinta, “Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems", Ann. Math. Studies 105, Princeton University Press, Princeton, 1983. | MR 717034 | Zbl 0516.49003

[12] M. Giaquinta, Growth conditions and regularity, a counterexample, Manuscripta Math. 59 (1987), 245-248. | MR 905200 | Zbl 0638.49005

[13] D. Gilbarg and N. S. Trudinger, “Elliptic Partial Differential Equations of Second Order", Grundlehren der math. Wiss. 224, second ed., revised third print., Springer, Berlin-Heidelberg-New York, 1998. | Zbl 0361.35003

[14] M. C. Hong, Some remarks on the minimizers of variational integrals with non standard growth conditions, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (7) 6-A (1992), 91-101. | MR 1164739 | Zbl 0768.49022

[15] P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions, Arch. Ration. Mech. Anal. 105 (1989), 267-284. | MR 969900 | Zbl 0667.49032

[16] P. Marcellini, Everywhere regularity for a class of elliptic systems without growth conditions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23 (1996), 1-25. | Numdam | MR 1401415 | Zbl 0922.35031

[17] U. Massari and M. Miranda, “Minimal Surfaces of Codimension One", North-Holland Mathematics Studies 91, North-Holland, Amsterdam-New York-Oxford, 1983. | MR 795963 | Zbl 0565.49030

[18] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinues, Ann. Inst. Fourier (Grenoble) 15.1 (1965), 189-258. | Numdam | MR 192177 | Zbl 0151.15401

[19] N. N. Ural'Tseva and A. B. Urdaletova, The boundedness of the gradients of generalized solutions of degenerate quasilinear nonuniformly elliptic equations, Vestn. Leningr. Univ. Mat. Mekh. Astron. 4 (1983), 50-56 (in Russian); English translation: Vestn. Leningr. Univ. Math. 16 (1984), 263-270. | MR 725829 | Zbl 0569.35029