×

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

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
26B25 Convexity of real functions of several variables, generalizations
35B65 Smoothness and regularity of solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[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
[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 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: 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: Springer Verlag Heidelberg
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.