\({\mathcal{C}^{\alpha}}\)-regularity for a class of non-diagonal elliptic systems with p-growth

  • Miroslav Bulíček
  • Jens Frehse


We consider weak solutions to nonlinear elliptic systems in a W 1,p -setting which arise as Euler equations to certain variational problems. The solutions are assumed to be stationary in the sense that the differential of the variational integral vanishes with respect to variations of the dependent and independent variables. We impose new structure conditions on the coefficients which yield everywhere \({\mathcal{C}^{\alpha}}\)-regularity and global \({\mathcal{C}^{\alpha}}\)-estimates for the solutions. These structure conditions cover variational integrals like \({\int F(\nabla u)\; dx}\) with potential \({F(\nabla u):=\tilde F (Q_1(\nabla u),\ldots, Q_N(\nabla u))}\) and positively definite quadratic forms in \({\nabla u}\) defined as \({Q_i(\nabla u)=\sum_{\alpha \beta} a_i^{\alpha \beta} \nabla u^\alpha \cdot \nabla u^\beta}\). A simple example consists in \({\tilde F(\xi_1,\xi_2):= |\xi_1|^{\frac{p}{2}} + |\xi_2|^{\frac{p}{2}}}\) or \({\tilde F(\xi_1,\xi_2):= |\xi_1|^{\frac{p}{4}}|\xi_2|^{\frac{p}{4}}}\). Since the Q i need not to be linearly dependent our result covers a class of nondiagonal, possibly nonmonotone elliptic systems. The proof uses a new weighted norm technique with singular weights in an L p -setting.

Mathematics Subject Classification (2000)

35J60 49N60 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bensoussan, A., Frehse, J.: Regularity results for nonlinear elliptic systems and applications. In: Applied Mathematical Sciences, vol. 151. Springer, Berlin (2002)Google Scholar
  2. 2.
    De Giorgi E.: Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. 3(3), 25–43 (1957)MathSciNetGoogle Scholar
  3. 3.
    De Giorgi E.: Un esempio di estremali discontinue per un problema variazionale di tipo ellittico. Boll. Un. Mat. Ital. 1(4), 135–137 (1968)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Eshelby J.D.: The force on an elastic singularity. Phil. Trans. R. Soc. Lond. A 244, 84–112 (1951)MathSciNetGoogle Scholar
  5. 5.
    Evans L.C.: Partial regularity for stationary harmonic maps into spheres. Arch. Ration. Mech. Anal. 116(2), 101–113 (1991). doi: 10.1007/BF00375587 CrossRefzbMATHGoogle Scholar
  6. 6.
    Fuchs M.: Topics in the Calculus of Variations. Advanced Lectures in Mathematics. Friedr. Vieweg & Sohn, Braunschweig (1994)Google Scholar
  7. 7.
    Giaquinta, M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems. In: Annals of Mathematics Studies, vol. 105. Princeton University Press, Princeton, NJ (1983)Google Scholar
  8. 8.
    Giaquinta, M., Hildebrandt, S.: Calculus of variations. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 310. Springer, Berlin (1996). The Lagrangian formalismGoogle Scholar
  9. 9.
    Hildebrandt S., Widman K.O.: Variational inequalities for vector-valued functions. J. Reine Angew. Math. 309, 191–220 (1979)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Knops R.J., Stuart C.A.: Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity. Arch. Ration. Mech. Anal. 86(3), 233–249 (1984). doi: 10.1007/BF00281557 CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Ladyzhenskaya, O.A., Uraltseva, N.N.: Linear and quasilinear elliptic equations. Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis. Academic Press, New York (1968)Google Scholar
  12. 12.
    Mingione G.: Regularity of minima: an invitation to the dark side of the calculus of variations. Appl. Math. 51(4), 355–426 (2006). doi: 10.1007/s10778-006-0110-3 CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Nash J.: Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80, 931–954 (1958)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Nečas, J.: Example of an irregular solution to a nonlinear elliptic system with analytic coefficients and conditions for regularity. In: Theory of Nonlinear Operators. Proceedings of the fourth Internat. Summer School, Acadamic Science, Berlin (1975)Google Scholar
  15. 15.
    Noether, E.: Invariant variation problems. Transp. Theory Stat. Phys. 1(3), 186–207 (1971). Translated from the German (Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, pp. 235–257 (1918)Google Scholar
  16. 16.
    Rajagopal K.R.: On implicit constitutive theories. Appl. Math. 48(4), 279–319 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Rajagopal K.R.: The elasticity of elasticity. Z. Angew. Math. Phys. 58(2), 309–317 (2007). doi: 10.1007/s00033-006-6084-5 CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Rajagopal K.R., Srinivasa A.R.: On the response of non-dissipative solids. Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 463, 2078 (2007). doi: 10.1098/rspa.2006.1760 CrossRefMathSciNetGoogle Scholar
  19. 19.
    Steffen, K.: Parametric surfaces of prescribed mean curvature. In: Calculus of Variations and Geometric Evolution Problems (Cetraro, 1996). LNCS, vol. 1713, pp. 211–265. Springer, Berlin (1999). doi: 10.1007/BFb0092671
  20. 20.
    Trautman A.: Noether equations and conservation laws. Comm. Math. Phys. 6, 248–261 (1967)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Uhlenbeck K.: Regularity for a class of non-linear elliptic systems. Acta Math. 138(3–4), 219–240 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Šverák V., Yan X.: Non-Lipschitz minimizers of smooth uniformly convex functionals. Proc. Natl. Acad. Sci. USA 99(24), 15269–15276 (2002). doi: 10.1073/pnas.222494699 CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Faculty of Mathematics and Physics, Mathematical InstituteCharles UniversityPraha 8Czech Republic
  2. 2.Department of Applied Analysis, Institute for Applied MathematicsUniversity of BonnBonnGermany

Personalised recommendations