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Regularity of minimizers of integrals of the calculus of variations with non-standard growth conditions. (English) Zbl 0667.49032

Let \(f(\xi)\in C^ 2(R^ n)\) be a real-valued strictly convex function such that \[ (1)\quad m| \xi |^ p\leq f(\xi)\leq M(1+| \xi |^ q)\quad for\quad all\quad \xi \in R^ n. \] Existence and regularity of a minimizer of the functional \[ F(u)=\int_{\Omega}f(Du(x))dx \] have been investigated when \(p=q\) and some suitable condition on the growth of the second derivatives of f are given.
Recently the interest on the regularity of minimizers of such functionals where \(p\neq q\) has been pointed out also in connection with some problems in nonlinear elasticity.
In this paper the author proposes an approach to the local regularity when the condition (1) is satisfied with \(p\neq q\) and also the second derivatives of f satisfy a non-standard condition.
Reviewer: R.Schianchi

MSC:

49S05 Variational principles of physics
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[1] E. Acerbi & N. Fusco, A regularity theorem for minimizers of quasiconvex integrals, Arch. Rational Mech. Analysis 99 (1987), 261–281. · Zbl 0627.49007
[2] H. Brezis, Analyse fonctionnelle, Théorie et applications, Masson, 1983.
[3] E. de Giorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino, Cl. Sci. Fis. Mat. Nat. 3 (1957), 25–43. · Zbl 0084.31901
[4] M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Math. Studies 105, Princeton Univ. Press, 1983. · Zbl 0516.49003
[5] M. Giaquinta, Growth conditions and regularity, a counterexample, Preprint Univ. Bonn, July 1987, to appear in Manuscripta Math. · Zbl 0638.49005
[6] M. Giaquinta & E. Giusti, On the regularity of minima of variational integrals, Acta Math. 148 (1982), 31–46. · Zbl 0494.49031
[7] D. Gilbarg, & N. S. Trudinger, Elliptic partial differential equations of second order, Grundl. Math. Wiss. 224, Springer-Verlag, 1977. · Zbl 0361.35003
[8] E. Giusti, Equazioni ellittiche del secondo ordine, Quaderni UMI n. 6, Pitagora Ed., 1978.
[9] O. Ladyzhenskaya & N. Ural’tseva, Linear and quasilinear elliptic equations, Math, in Science and Engineering 46, Academic Press, 1968.
[10] P. Marcellini, Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals, Manuscripta Math. 51 (1985), 1–28. · Zbl 0573.49010
[11] P. Marcellini, On the definition and the lower semicontinuity of certain quasiconvex integrals, Ann. Inst. Henri Poincaré, Analyse non linéaire 3 (1986), 391–409. · Zbl 0609.49009
[12] P. Marcellini, Un exemple de solution discontinue d’un problème variationnel dans le cas scalaire, Manuscript, September 1987.
[13] C. B. Morrey, Multiple integrals in the calculus of variations, Grundl. Math. Wiss. 130, Springer-Verlag, 1966. · Zbl 0142.38701
[14] J. Moser, A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations, Communications Pure Appl. Math. 13 (1960), 457–468. · Zbl 0111.09301
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