Fan, Xianling; Zhao, Dun A class of De Giorgi type and Hölder continuity. (English) Zbl 0927.46022 Nonlinear Anal., Theory Methods Appl. 36, No. 3, A, 295-318 (1999). Investigating the regularity of solutions of quasilinear elliptic equations with the standard \(m\)-growth condition one can use the function class \({\mathcal B}_m\), which was introduced by Ladyzhenskaya and Ural’tseva. The authors are interested in \(m(x)\)-growth conditions with \(m(x)\) being a positive, continuous, bounded function. The last conditions are of the form \[ c_1| z|^{m(x)}- c_0\leq F(x,z)\leq c_2| z|^{m(x)}- c_0, \] with \(F: \Omega\times \mathbb{R}^n\to \mathbb{R}\), \(\Omega\subset \mathbb{R}^n\), and \(1< p\leq m(x)\leq q\) for \(x\in\Omega\).To study the regularity of the corresponding quasilinear equations, they introduce a class \({\mathcal B}_{m(x)}\) which is a generalization of the class \({\mathcal B}_m\). It is proved that if the function \(m(x)\) satisfies some additional conditions (in particular \(m(x)\) is Hölder continuous) then the functions belonging to \({\mathcal B}_{m(x)}\) are Hölder continuous. In consequence the Hölder continuity of minimizers of variational functionals satisfying \(m(x)\) growth conditions follows. As application the Hölder regularity of solutions of quasilinear equations with principal part in divergence form and with the same growth conditions can be proved. Reviewer: L.Skrzypczak (Poznań) Cited in 2 ReviewsCited in 182 Documents MSC: 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 49N60 Regularity of solutions in optimal control 47F05 General theory of partial differential operators Keywords:quasilinear elliptic equations; Hölder regularity; solutions of quasilinear equations; principal part in divergence form PDFBibTeX XMLCite \textit{X. Fan} and \textit{D. Zhao}, Nonlinear Anal., Theory Methods Appl. 36, No. 3, 295--318 (1999; Zbl 0927.46022) Full Text: DOI References: [1] Acerbi, E.; Fusco, N., Partial regularity under anisotropic \((p,q)\) growth conditions, J. Differential Equations, 107, 46-67 (1994) · Zbl 0807.49010 [2] Acerbi, E.; Fusco, N., A transmission problem in the calculus of variations, Calc. Var., 2, 1-16 (1994) · Zbl 0791.49041 [3] E. De Giorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino 3 (3) (1957) 25-43.; E. De Giorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino 3 (3) (1957) 25-43. [4] E. Di Benedetto, \(C^{1+α}\); E. Di Benedetto, \(C^{1+α}\) [5] E. Di Benedetto, N.S. Trudinger, Harnack inequality for quasiminima of variational integrals, Ann. Inst. H. Poincare: Anal. Nonlineaire 1 (1984) 295-308.; E. Di Benedetto, N.S. Trudinger, Harnack inequality for quasiminima of variational integrals, Ann. Inst. H. Poincare: Anal. Nonlineaire 1 (1984) 295-308. [6] Donaldson, T., Nonlinear elliptic boundary value problems in Orlicz-Sobolev space, J. Differential Equations, 10, 507-528 (1971) · Zbl 0218.35028 [7] Evans, L. C., A new proof of local \(C^{1,α}\) regularity for solutions of certain degenerate elliptic p.d.e., J. Differential Equations, 45, 356-373 (1982) · Zbl 0508.35036 [8] Fusco, N.; Sbordone, C., Some remarks on the regularity of minima of anisotropic integrals, Comm. Partial Differential Equations, 18, 153-167 (1993) · Zbl 0795.49025 [9] Giaquinta, M., Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems (1983), Princeton University Press: Princeton University Press Princeton · Zbl 0516.49003 [10] Giaquinta, M., Growth conditions and regularity, a counterexample, Manuscr. Math., 59, 245-248 (1987) · Zbl 0638.49005 [11] Giaquinta, M.; Giusti, E., On the regularity of the minima of variational integrals, Acta Math., 148, 31-46 (1982) · Zbl 0494.49031 [12] Hong, M.-C., Some remarks on the minimizers of variational integrals with nonstandard growth conditions, Boll. Unione Mat. Ital. Ser. A, 6, 7, 91-102 (1992) [13] O.A. Ladyzhenskaya, Ural’tseva, N.N., Linear and Quasilinear Elliptic Equations, 2nd Russian ed., Nauka, Moscow, 1973.; O.A. Ladyzhenskaya, Ural’tseva, N.N., Linear and Quasilinear Elliptic Equations, 2nd Russian ed., Nauka, Moscow, 1973. [14] Marcellini, P., Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch. Rat. Mech. Anal., 105, 267-284 (1989) · Zbl 0667.49032 [15] Marcellini, P., Regularity and existence of solutions of elliptic equations with \(p,q\)-growth conditions, J. Differential Equations, 90, 1-30 (1991) · Zbl 0724.35043 [16] Morrey, C. B., Multiple Integrals in the Calculus of Variations (1968), Springer: Springer Berlin [17] J.M. Rakotoson, R. Temam, Relative rearrangement in quasilinear variational elliptic inequalities, Indiana Univ. Math. J. 36 (1987).; J.M. Rakotoson, R. Temam, Relative rearrangement in quasilinear variational elliptic inequalities, Indiana Univ. Math. J. 36 (1987). [18] Uhlenbeck, K., Regularity for a class of non-linear elliptic systems, Acta Math., 138, 219-240 (1977) · Zbl 0372.35030 [19] Zhikov, V. V., Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR Izv., 29, 33-36 (1987) · Zbl 0599.49031 [20] Zhikov, V. V., On passing to the limit in nonlinear variational problem, Mat. Sbornik, 183, 8, 47-84 (1992) · Zbl 0767.35021 [21] Ziemer, W. P., Boundary regularity for quasiminima, Arch. Rat. Mech. Anal., 92, 4, 371-382 (1986) · Zbl 0611.35030 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.