Annali di Matematica Pura ed Applicata

, Volume 188, Issue 1, pp 61–122

Partial regularity and singular sets of solutions of higher order parabolic systems

Article

Abstract

In the present paper we provide a broad survey of the regularity theory for non-differentiable higher order parabolic systems of the type
$$ \int \limits_{\Omega_T} u\cdot \varphi_t - A(z,u,Du,\dots,D^m u) \cdot D^m \varphi \, {\rm d}z =\int \limits_{\Omega_T} \sum_{k=0}^{m-1} B^k(z,u,Du,\dots,D^m u) \cdot D^k\varphi \, {\rm d}z.$$
Initially, we prove a partial regularity result with the method of A-polycaloric approximation, which is a parabolic analogue of the harmonic approximation lemma of De Giorgi. Moreover, we prove better estimates for the maximal parabolic Hausdorff-dimension of the singular set of weak solutions, using fractional parabolic Sobolev spaces. Thereby, we also consider different situations, which yield a better dimension reduction result, including the low dimensional case and coefficients A(z, Dmu), independent of the lower order derivatives of u.

Keywords

Partial regularity Singular set Higher order parabolic systems 

Mathematics Subject Classification (2000)

35D10 35G20 35K55 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Acerbi E. and Mingione G. (2001). Regularity results for a class of functionals with non-standard growth. Arch. Rat. Mech. Anal. 156: 121–140 MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Acerbi E. and Mingione G. (2007). Gradient estimates for a class of parabolic systems. Duke Math. J. 136: 285–320 MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Acerbi E., Mingione G. and Seregin G.A. (2004). Regularity results for parabolic systems related to a class of non-newtonian fluids. Ann. Inst. Henri Poincaré Anal. Non Linéaire 21(1): 25–60 MATHMathSciNetGoogle Scholar
  4. 4.
    Adams R.A. (1978). Sobolev Spaces. Academic Press, New York Google Scholar
  5. 5.
    Bögelein, V.: Regularity results for weak and very weak solutions of higher order parabolic systems. Ph.D.Thesis (2007)Google Scholar
  6. 6.
    Bojarski B. and Iwaniec T. (1983).Analytical foundations of the theory of quasiconformal mappings in \({\mathbb R^n}\). Ann. Acad. Sci. Fenn. Ser. A I 8: 257–324 MATHMathSciNetGoogle Scholar
  7. 7.
    Campanato S. (1966). Equazioni paraboliche del secondo ordine e spazi \({\fancyscript {L}^{2,\theta} (\Omega,\delta)}\). Ann. Mat. Pura Appl. IV Ser. 73: 55–102 MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Campanato S. (1967). Maggiorazioni interpolatorie negli spazi \({H_\lambda^{m,p}(\Omega)}\). Ann. Mat. Pura Appl. 75: 261–276 CrossRefMathSciNetGoogle Scholar
  9. 9.
    Campanato S. (1982). Differentiability of the solutions of nonlinear elliptic systems with natural growth. Ann. Mat. Pura Appl. 131(4): 75–106 MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Campanato S. (1984). On the nonlinear parabolic systems in divergence form. Hölder continuity and partial Hölder continuity of the solutions. Ann. Mat. Pura Appl. 137(4): 83–122 MATHMathSciNetGoogle Scholar
  11. 11.
    Campanato S. and Cannarsa P. (1980). Differentiability and partial Hölder continuity of the solutions of non-linear elliptic systems of order 2m with quadratic growth. Ann. Scuola Norm. Sup. Pisa 8: 285–309 MathSciNetGoogle Scholar
  12. 12.
    Da Prato G. (1965). Spazi \({\fancyscript {L}^{(p,\vartheta)}(\Omega,\delta)}\) e loro proprieta. Ann. Mat. Pura Appl. IV. Ser. 69: 383–392 MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    De Giorgi, E.: Frontiere orientate di misura minima. Sem. Scuola Normale Superiore Pisa (1960–1961)Google Scholar
  14. 14.
    Domokos A. (2004). Differentiability of solutions for the non-degenerate p-laplacian in the heisenberg group. J. Differ. Equ. 204: 439–470 MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Duzaar F., Gastel A. and Grotowski J.F. (2001). Optimal partial regularity for nonlinear elliptic systems of higher order. J. Math. Sci., Tokyo 8(3): 463–499 MATHMathSciNetGoogle Scholar
  16. 16.
    Duzaar F. and Grotowski J.F. (2000). Optimal interior partial regularity for nonlinear elliptic systems: the method of a-harmonic approximation. Manuscr. Math. 103: 267–298 MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Duzaar F. and Mingione G. (2004). The p-harmonic approximation and the regularity of p-harmonic maps. Calc. Var. Part. Differ. Equ. 20: 235–256 MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Duzaar F. and Mingione G. (2004). Regularity for degenerate elliptic problems via p-harmonic approximation. Ann. Inst. Henri Poincaré Anal. Non Linéaire 21: 735–766 MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Duzaar F. and Mingione G. (2005). Second order parabolic systems, optimal regularity and singular sets of solutions. Ann. Inst. Henri Poincaré Anal. Non Linéaire 22: 705–751 MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Duzaar, F., Mingione, G., Steffen, K.: Second order parabolic systems with p-growth and regularity (to appear, 2008)Google Scholar
  21. 21.
    Fefferman C. and Stein E.M. (1972). Hp spaces of several variables. Acta Math. 129: 137–193 MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Föglein, A.: Regularität von Lösungen gewisser Systeme elliptischer partieller Differentialgleichungen in der Heisenbarg-Gruppe. Diplomarbeit (2005)Google Scholar
  23. 23.
    Giaquinta M. (1983). Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton University Press, Princeton MATHGoogle Scholar
  24. 24.
    Giaquinta, M.: Introduction to regularity theory for nonlinear elliptic systems (1993)Google Scholar
  25. 25.
    Giaquinta M. and Struwe M. (1982). On the partial regularity of weak solutions of nonlinear parabolic systems. Math. Z. 179: 437–451 MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Giusti E. (2003). Direct Methods in the Calculus f Variations. World Scientific Publishing Company, Tuck Link, Singapore Google Scholar
  27. 27.
    John J. and Nirenberg L. (1961). On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14: 415–426 MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Kinnunen J. and Lewis J.L. (2000). Higher integrability for parabolic systems of p-laplacian type. Duke Math. J. 102: 253–271 MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Ladyženskaja O.A., Solonnikov V.A. and Ural’ceva N.N. (1968). Linear and Quasilinear Equations of Parabolic Type, vol. 23. American Mathematical Society, Providence Google Scholar
  30. 30.
    Mingione G. (2003). Bounds for the singular set of solutions to non linear elliptic systems. Calc. Var. Partial Differ. Equ. 18(4): 373–400 MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Mingione G. (2003). The singular set of solutions to non-differentiable elliptic systems. Arch. Ration. Mech. Anal. 166: 287–301 MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Nirenberg L. (1960). On elliptic partial differential equations. Ann. Sc. Norm. Super., Pisa III. Ser. 123: 115–162 Google Scholar
  33. 33.
    Simon J. (1987). Compact sets in the space L p(0, T; B). Ann. Mat. Pura Appl. IV. Ser. 146: 65–96 MATHCrossRefGoogle Scholar
  34. 34.
    Simon, L.: Lectures on Geometric Measure Theory. Proc. Centre Math. Anal., Austr. Nat. Univ., Canberra (1983)Google Scholar
  35. 35.
    Simon, L.: Theorems on Regularity and Singularity of Energy Minimizing Maps. Lectures in Math., ETH Zrich, Birkhuser, Basel (1996)Google Scholar
  36. 36.
    Stredulinsky E.W. (1980). Higher integrability from reverse Hölder inequalities. Indiana Univ. Math. J. 29: 407–413 MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department MathematikUniversität Erlangen–NürnbergErlangenGermany

Personalised recommendations