Interior \(C^{1,\gamma}\)-regularity for weak solutions of nonlinear second order elliptic systems. (English) Zbl 1174.35358

Summary: The interior \(C^{0,\gamma}\)-regularity for the first gradient of a weak solution to a class of nonlinear second order elliptic systems is proved under the assumption that oscillations of coefficients are controlled by the ellipticity constant.


35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35B65 Smoothness and regularity of solutions to PDEs
35D10 Regularity of generalized solutions of PDE (MSC2000)
Full Text: DOI


[1] S. Campanato Sistemi ellittici in forma divergenza. Regolarita all’interno, Quaderni Scuola Norm. Sup. Pisa (Pisa, 1980). · Zbl 0453.35026
[2] Campanato, A maximum principle for non-linear elliptic systems: Boundary fundamental estimates, Advances Math. 66 pp 293– (1987) · Zbl 0644.35042
[3] Daněček, On the regularity of weak solutions to nonlinear elliptic systems of second order, Z. Anal. Anwendungen 9 pp 535– (6) · Zbl 0735.35034 · doi:10.4171/ZAA/422
[4] Daněček, On the , n-regularity of the gradient of weak solution to the nonlinear elliptic systems, Comment. Math. Univ. Carolinae 37 (3) pp 523– (1996)
[5] Daněček, The example of a nonlinear second order elliptic systems in three dimensions, to appear in Comment. Math. Univ. Carolinae 45 (2004)
[6] Evans, Linearisation at infinity and Lipschitz estimates for certain problems in the calculus of variation, Proceedings of the Royal Society of Edinburgh 102A pp 291– (1986)
[7] M. Giaquinta Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies 105 (Princenton University Press, Princeton, 1983). · Zbl 0516.49003
[8] Giaquinta, On the regularity of weak solutions to nonlinear elliptic systems via Liouville’s property, Comment. Math. Univ. Carolinae 20 pp 111– (1979) · Zbl 0396.35047
[9] Giaquinta, On the regularity of weak solutions to nonlinear elliptic systems, J. Reine Angew. Math. 316 pp 140– (1981) · Zbl 0438.35019
[10] Grevholm, On the structure of the spaces, {\(\lambda\)}k, Math. Scand. 26 pp 189– (1970) · Zbl 0212.46002
[11] Hao, An example of irregular solutions to a nonlinear Euler-Lagrange system with real analytic coefficients, Ann. Scuola Norm. Sup. Pisa 23 pp 57– (1996) · Zbl 0864.35031
[12] A. I. Koshelev Regularity problem for quasilinear elliptic and parabolic systems, Lecture Notes in Mathematics 1614 (Springer-Verlag Heidelberg, 1995). · Zbl 0847.35023 · doi:10.1007/BFb0094482
[13] A. Kufner O. John S. Fučík Function Spaces (Academia, Prague, 1977).
[14] O. A. Ladyzhenskaya N. N. Uralceva Linear and quasilinear elliptic equations (Nauka, Moskva, 1973).
[15] S. Leonardi Remarks on the regularity of solutions of elliptic systems, in: Applied Nonlinear Analysis, edited by A. Sequeira, H. B. da Veiga, and J. H. Videman (Kluwer Academic/Plenum Publishers, New York, 1999), pp. 325-344. · Zbl 0952.35034
[16] Leonardi, On constants of some regularity theorems. De Giorgi’s type counterexample, Mathematische Nachrichten 192 pp 191– (1998) · Zbl 0909.35030
[17] Mazya, Examples of nonregular solutions of quasilinear elliptic equations with analytic coefficients, Funktsionalnyi analiz i ego prilozhenia 2 (1968)
[18] Morrey, On the solutions of quasilinear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 pp 126– (1938)
[19] J. Nečas Introduction to the Theory of Nonlinear Elliptic Equations, Teubner-Texte zur Mathematik 52 (Teubner, Leipzig, 1983).
[20] Stará, Regularity result for non-linear elliptic systems in two dimensions, Ann. Scuola Norm. Sup. Pisa 25 pp 163– (1971) · Zbl 0215.45601
[21] Šverák, A singular minimizer of smooth strongly convex functional in three dimensions, Calc. Var. Partial Differential Equations 10 (3) pp 213– (2000) · Zbl 1013.49027 · doi:10.1007/s005260050151
[22] Šverák, Non-Lipschitz minimizers of smooth strongly convex functionals, Proc. Nat. Acad. Sci. USA 99 (24) pp 15269– (2002) · Zbl 1106.49046 · doi:10.1073/pnas.222494699
[23] W. P. Ziemer Weakly Differentiable Functions (Springer-Verlag, Heidelberg, 1989).
[24] Balanda, On Liouville theorem and the regularity of weak solutions to some nonlinear elliptic systems of higher order, Comment. Math. Univ. Carolinae 32 pp 615– (1991) · Zbl 0773.35017
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.