Fusco, Nicola; Sbordone, Carlo Higher integrability of the gradient of minimizers of functionals with nonstandard growth conditions. (English) Zbl 0727.49021 Commun. Pure Appl. Math. 43, No. 5, 673-683 (1990). In the paper, functionals of the form \(F(u,\Omega)=\int_{\Omega}f(| Du|)dx\) are considered, where \(\Omega\) is a bounded open subset of \({\mathbb{R}}^ n\), and u varies among all functions in \(W^{1,1}_{loc}(\Omega;{\mathbb{R}}^ m)\). A function \(u\in W^{1,1}_{loc}(\Omega;{\mathbb{R}}^ m)\) is called a local minimizer for F if for every function \(w\in W^{1,1}(\Omega;{\mathbb{R}}^ m)\) with compact support F(u,spt w)\(\leq F(u+w,spt w)\). The main result of the paper is the following higher integrability result. Assume that f: \([0,+\infty [\to [0,+\infty [\) is convex, increasing and that there exist \(1<p\leq q\) such that \(f(t)/t^ p\) is increasing, \(f(t)/t^ q\) is decreasing, and \(p>nq/(n+q)\). Then, if u is a local minimizer for F, there exists \(r>1\) such that \(f(| Du|)\in L^ r_{loc}(\Omega).\) The same result holds for functionals of the form \(\int_{\Omega}g(x,u,Du)dx\) provided there exists a function f with the properties above such that f(\(| z| (\leq g(x,s,z)\leq xc(1+f(| z|))\forall x,s,z\). Reviewer: G.Buttazzo (Pisa) Cited in 71 Documents MSC: 49N60 Regularity of solutions in optimal control 49J40 Variational inequalities Keywords:nonstandard growth conditions; local minimizer; higher integrability result × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Bojarski, Ann. Acad. Sci. Fen. 8 pp 257– (1983) · Zbl 0548.30016 · doi:10.5186/aasfm.1983.0806 [2] personal communication. [3] and , Higher integrability from reverse Jensen inequalities with different supports, in Partial Differential Equations and the Calculus of Variations: Essays in Honour of Ennio DeGiorgi, Prog. Nonlin. Diff. Eq. Appl., Birkhäuser Boston, 1989, pp. 541–561. [4] Gehring, Acta Math. 130 pp 265– (1973) [5] Giaquinta, Manuscripta Math. 59 pp 245– (1987) [6] Giaquinta, Acta Math. 148 pp 31– (1982) [7] and , Regularity results for some classes of higher-order nonlinear elliptic systems, J. Reine Angew. Math. 311–312, 1979, pp. 145–169. [8] , and , Inequalities, Cambridge University Press, 1952. [9] and , Convex Functions and Orlicz Spaces, Noordhoff Ltd., New York, 1961. · Zbl 0095.09103 [10] Une example de solution discontinue d’un problème variationnel dans le cas scalaire, preprint, 1987. [11] Marcellini, Arch. Rat. Mech. Anal. 105 pp 267– (1989) [12] Muckenhoupt, Trans. Amer. Math. Soc. 165 pp 207– (1972) [13] Sbordone, Boll. U.M.I. An. Funz. Appl., serie VI pp 73– (1986) 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.