Hölder continuity of minimizers of functionals with variable growth exponent. (English) Zbl 0878.49010

The paper deals with the Hölder continuity of local minimizers of integral functionals whose model is \[ F_0(u)= \int_\Omega|Du|^{a(x)}dx. \] Here \(\Omega\) is an open subset of \(\mathbb{R}^n\), the functions \(u\) are scalar, and the function \(a(x)\) is assumed to be in \(W^{1,s}(\Omega)\) with \(s>n\) and such that for a suitable \(p_0\in]1,n]\), \[ p_0\leq a(x)\leq p^*_0\quad\text{on }\Omega.\tag{1} \] The main result is that under these assumptions every local minimizer of \(F_0\) is Hölder continuous. Actually, the result proved holds for quasi minimizers of \(F_0\), and so the class of functionals for which the Hölder continuity result holds is considerably wider and includes integral functionals of the form \[ F(u)=\int_\Omega f(x,u,Du)dx, \] where the integrand \(f\) satisfies the growth assumption \[ c_1(|z|^{a(x)}-|s|^{r(x)}- 1)\leq f(x,s,z)\leq c_2(|z|^{a(x)}+|s|^{r(x)}+ 1) \] with \(a(x)\) verifying (1) and \(0\leq r(x)\leq p^*_0\) on \(\Omega\).
The reader can find a complete discussion about functionals with different growth from below and from above in the paper by P. Marcellini [J. Differ. Equations 105, No. 2, 296-333 (1993; Zbl 0812.35042)], where also a wide bibliography can be found.
Reviewer: G.Buttazzo (Pisa)


49J45 Methods involving semicontinuity and convergence; relaxation
49N60 Regularity of solutions in optimal control


Zbl 0812.35042
Full Text: DOI EuDML


[1] ACERBI E., FUSCO N.: Partial regularity under anisotropic (p, q) growth conditions.J. Diff. Equations. 107 (1994), 46–67. · Zbl 0807.49010
[2] ACERBI E., FUSCO N.: A transmission problem in the calculus of variations.Calc. Var. 2 (1994), 1–16. · Zbl 0791.49041
[3] ALKHUTOV YU. A.: The Hölder continuity of certain nonlinear elliptic equations with nonstandard growth condition.Differentsial’nye Uravneniya (1997) (to appear). · Zbl 0949.35048
[4] BOCCARDO, L., MARCELLINI, P., SBORDONE, C.: Lregularity for a variational problem with sharp non standard growth conditions.Boll. Un. Mat. Ital. (7)4-A (1990), 219–225. · Zbl 0711.49058
[5] BREZIS H.:Analyse fonctionnelle. Théorie et applications. Masson, Paris, 1983.
[6] DE GIORGI E.: Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari.Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 3 (1957), 25–43. · Zbl 0084.31901
[7] FAN X.: A class of De Giorgi type and Hölder continuity of minimizers of variationals withm(x)-growth condition. Preprint, Lanzhou University, China, 1995.
[8] FUSCO N., SBORDONE C.: Local boundedness of minimizers in a limit case.Manuscripta Math. 69 (1990), 19–25. · Zbl 0722.49012
[9] FUSCO N., SBORDONE C.: Some remarks on the regularity of minima of anisotropic integrals.Comm. Partial Differential Equations 18 (1993), 154–167. · Zbl 0795.49025
[10] GIAQUINTA M.: Growth conditions and regularity, a counterexample.Manuscripta Math. 59 (1987), 245–248. · Zbl 0638.49005
[11] GIAQUINTA M., GIUSTI E.: On the regularity of the minima of variational integrals.Acta Math. 148 (1982), 31–46. · Zbl 0494.49031
[12] GIAQUINTA M., GIUSTI E.: Quasi-minima.Ann. Inst. H. Poincaré, Analyse non linéaire.1 (1984), 79–107. · Zbl 0541.49008
[13] GIUSTI E.:Metodi diretti nel calcolo delle variazioni. Pitagora, Bologna, 1994. · Zbl 0942.49002
[14] HONG M. C.: Some remarks on the minimizers of variational integrals with non standard growth conditions.Boll. Un. Mat. Ital. (7)6-A (1992), 91–102. · Zbl 0768.49022
[15] LADYZHENSKAJA O. A., URAL’TSEVA N. N.:Equations aux dérivées partielles du type elliptique. Dunod, Paris, 1968.
[16] MARCELLINI P.: Un exemple de solution discontinue d’un problème variationnel dans le cas scalaire.Preprint Univ. Florence, (1987).
[17] MARCELLINI P.: Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions.Arch. Rational Mech. Anal. 105 (1989), 267–284. · Zbl 0667.49032
[18] MARCELLINI P.: Regularity and existence of solutions of elliptic equations withp,q-growth conditions.J. Diff. Equations. 90 (1991), 1–30. · Zbl 0724.35043
[19] MARCELLINI P.: Regularity for elliptic equations with general growth conditions.J. Diff. Equations. 105 (1993), 296–333. · Zbl 0812.35042
[20] MARCUS M., MIZEL V. J.: Absolute continuity on tracks and mappings of Sobolev Spaces.Arch. Rational Mech. Anal. 45 (1972), 294–320. · Zbl 0236.46033
[21] MASCOLO E., PAPI G.: Local boundedness of minimizers of integrals of the calculus of variations.Ann. Mat. Pura Appl. 167 (1994), 323–339. · Zbl 0819.49023
[22] MORREY C. B.:Multiple integrals in the calculus of variations. Springer, Berlin, 1968. · Zbl 0142.38701
[23] MOSCARIELLO G., NANIA L.: Hölder continuity of minimizers of functionals with non standard growth conditions.Ricerche Mat. XL (1991), 259–273. · Zbl 0773.49019
[24] ZHIKOV V. V.: Averaging of functionals of the calculus of variations and elasticity theory.Math. USSR Izvestiya.29 (1987), 33–66. · Zbl 0599.49031
[25] ZHIKOV V. V.: Lavrentiev phenomenon and homogenization for some variational problems.Proc. Workshop ”Composite media and homogenization theory”, Trieste, 1995. · Zbl 0783.35005
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