Mascolo, Elvira; Papi, Gloria Harnack inequality for minimizers of integral functionals with general growth conditions. (English) Zbl 0855.49027 NoDEA, Nonlinear Differ. Equ. Appl. 3, No. 2, 231-244 (1996). In this paper the authors prove the Harnack inequality for minimizers of integral functionals of the calculus of variations of the type \[ \int_\Omega f \bigl( |Du |\bigr) dx. \] They assume that \(f\) satisfies some conditions which allow nonstandard growth such as \[ c_1 t^p - c_2 \leq f(t) \leq c_3 t^m + c_4 \] with \(c_i \in R_+\) and \(1 < p < m\). Reviewer: R.Schianchi (Roma) Cited in 19 Documents MSC: 49N60 Regularity of solutions in optimal control 49J45 Methods involving semicontinuity and convergence; relaxation Keywords:nonstandard growth condition; Harnack inequality; minimizers; integral functionals × Cite Format Result Cite Review PDF Full Text: DOI References: [1] R.A. ADAMS,Sobolev spaces, Academic Press, New York, 1975 [2] T. BHATTACHARAYA, F. LEONETTI,W 2,2 regularity for weak solutions of elliptic systems with non standard growth,J. Math. Anal. Appl.,176(1), 224-234 (1993) · Zbl 0809.35008 · doi:10.1006/jmaa.1993.1210 [3] E. DI BENEDETTO, N.S. TRUDINGER, Harnack inequality for quasiminima of variational integrals,Ann. Inst. H. Poincaré, Anal. Non Linéare 1(4), 295-308 (1984) [4] N. FUSCO, C. SBORDONE, Some remarks on the regularity of minima of anisotropic integrals,Comm. Partial Diff. Equ.,18, 158-167 (1993) · Zbl 0795.49025 · doi:10.1080/03605309308820924 [5] M. GIAQUINTA, E. GIUSTI, On the regularity of minima of variational integrals,Acta Math. 148, 31-46 (1982) · Zbl 0494.49031 · doi:10.1007/BF02392725 [6] M. GIAQUINTA, Multiple integrals in the calculus of variations and nonlinear elliptic systems,Annals of math. Studies 105, Princeton Univ. Press, Princeton, 1983 · Zbl 0516.49003 [7] E. GIUSTI,Equazioni ellittiche del secondo ordine, Quaderni U.M.I.6, Pitagora, Bologna, 1978 [8] E. GIUSTI,Metodi diretti nel calcolo delle variazioni U.M.I., Bologna, 1994 [9] N.V. KRYLOV, M.V. SAFONOV, Certain properties of solutions of parabolic equations with measurable coefficients,Izvestia Akad. Nauk. SSSR, t40, 161-175 (1980),English Transl. Math. USSR Izv., t.16, (1981) [10] M.A. KRASNOSEL’SKII, Y.A. RUTICKII,Convex functions and Orlicz spaces, Noordhoff Ltd, Groningen, 1961 [11] O.A. LADY?ENSKAJA, N.N. URAL’CEVA,Linear and quasilinear elliptic equations, Math. in Sciences and Engineering46, Academic Press, 1968 [12] G.M. LIEBERMAN, The natural generalization of the natural conditions of Lady?enskaja and Ural’ceva for elliptic equations,Commun. Partial Diff. Equ.,16, 311-361 (1991) · Zbl 0742.35028 · doi:10.1080/03605309108820761 [13] P. MARCELLINI, Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions,Arch. Rat. Mech. Anal. 105(5), 267-284 (1989) · Zbl 0667.49032 · doi:10.1007/BF00251503 [14] P. MARCELLINI, Regularity and existence of solutions of elliptic equations withp,q-growth conditions.J. Differential Equations 90, 1-30 (1991) · Zbl 0724.35043 · doi:10.1016/0022-0396(91)90158-6 [15] P. MARCELLINI, Regularity for elliptic equations with general growth conditions,J. Differential Equations 105, 296-333 (1993) · Zbl 0812.35042 · doi:10.1006/jdeq.1993.1091 [16] P. MARCELLINI, Everywhere regularity for a class of elliptic systems without growth conditions,J. Differential Equations, to appear · Zbl 0922.35031 [17] E. MASCOLO, G. PAPI, Local boundedness of minimizers of integrals of the calculus of variations,Annali Mat. Pura Appl. 167, 323-339 (1994) · Zbl 0819.49023 · doi:10.1007/BF01760338 [18] G. MOSCARIELLO, Regularity results for quasi minima of functionals with nopolynomial growth,J. Math. Anal. Appl. 168, 500-512 (1992) · Zbl 0768.49017 · doi:10.1016/0022-247X(92)90175-D [19] G. MOSCARIELLO, L. NANIA, Hölder continuity of minimizers of functionals with non standard growth conditions,Ricerche di Matematica,15(2), 259-273 (1991) · Zbl 0773.49019 [20] M.M. RAO, Z.D. REN,Theory of Orlicz spaces, Marcel Dekker, New York, 1991 · Zbl 0724.46032 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.