Partial regularity of minimizers of quasiconvex integrals. (English) Zbl 0594.49004

The authors consider variational integrals \(\int_{\Omega}f(x,u,Du)dx\) with f(x,u,p) growing polynomially, of class \(C^ 2\) in p and Hölder continuous in (x,u). Under the main assumption that f(x,u,p) is uniformly strictly quasiconvex they prove that each minimizer is of class \(C^{1,\mu}\) in an open set \(\Omega_ 0\subset \Omega\) such that \(meas(\Omega -\Omega_ 0)=0\).
Reviewer: R.Schianchi


49J45 Methods involving semicontinuity and convergence; relaxation
26B25 Convexity of real functions of several variables, generalizations
35B65 Smoothness and regularity of solutions to PDEs
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[1] Acerbi, E.; Fusco, N., Semicontinuity problems in the calculus of variations, Arch. Rat. Mech. Anal., t. 86, 125-145, (1984) · Zbl 0565.49010
[2] Ball, J. M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal., t. 63, 337-403, (1977) · Zbl 0368.73040
[3] Coiffman, R. R.; Fefferman, C., Weighted norm inequalities for maximal functions and singular integrals, Studia Math., t. 51, 241-250, (1974) · Zbl 0291.44007
[4] de Giorgi, E., Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino, cl. Sci. Fis. Mat. Nat., 3, t. 3, 25-43, (1957) · Zbl 0084.31901
[5] E. Di Benedetto, T. S. Trudinger, Harnack inequalities for quasi-minima of variational integrals. Ann. Inst. H. Poincaré. Analyse non linéaire.
[6] C. L. Evans, Quasiconvexity and partial regularity in the calculus of variations. Preprint. · Zbl 0623.49008
[7] Gehring, F. W., The L^{p} integrability of the partial derivatives of a quasi conformal mapping, Acta Math., t. 130, 265-277, (1973) · Zbl 0258.30021
[8] Giaquinta, M., Multiple integrals in the calculus of variations and nonlinear elliptic systems, (1983), Princeton Univ. Press Princeton · Zbl 0516.49003
[9] Giaquinta, M.; Giusti, E., On the regularity of minima of variational integrals, Acta Math., t. 148, 31-46, (1982) · Zbl 0494.49031
[10] Giaquinta, M.; Giusti, E., Differentiability of minima of non-differentiable functionals, Inventiones Math., t. 72, 285-298, (1983) · Zbl 0513.49003
[11] Giaquinta, M.; Giusti, E., Quasi-minima, Ann. Inst. H. Poincaré, Analyse non linéaire, t. 1, 79-107, (1984) · Zbl 0541.49008
[12] Giaquinta, M.; Giusti, E., Sharp estimates for the derivatives of local minima of variational integrals, Boll. U. M. I., 6, t. 3-A, 239-248, (1984) · Zbl 0543.49019
[13] Giaquinta, M.; Ivert, P. A., Partial regularity for minima of variational integrals, (1984), Zürich ETH, Preprint FIM
[14] Giaquinta, M.; Modica, G., Regularity results for some classes of higher order nonlinear elliptic systems, J. reine u. angew. Math., t. 311-312, 145-169, (1979) · Zbl 0409.35015
[15] Giaquinta, M.; Modica, G., Almost-everywhere regularity results for solutions of nonlinear elliptic systems, Manuscripta math., t. 28, 109-158, (1979) · Zbl 0411.35018
[16] Giaquinta, M.; Soucek, J., Caccioppoli’s inequality and Legendre-Hadamard condition, Math. Ann., t. 270, 105-107, (1985) · Zbl 0561.35027
[17] Morrey, C. B., Quasi convexity and the lower semicontinuity of multiple integrals, Pacific J. Math., t. 2, 25-53, (1952) · Zbl 0046.10803
[18] Morrey, jr, C. B., Multiple integrals in the calculus of variations, (1966), Springer Verlag Heidelberg
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