Second order parabolic systems, optimal regularity, and singular sets of solutions;Systèmes paraboliques de la deuxième ordre, regularité optimale, et des ensembles singuliers des solutions

Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

Volumn 22, Issue 6, 2005, Pages 705-751

  Duzaar, Frank ^a   Mingione, Giuseppe ^b  

^a UNIVERSITY OF ERLANGEN NUREMBERG   (Germany)

^b UNIVERSITY OF PARMA   (Italy)

Author keywords

Parabolic systems; Partial regularity; Singular sets

Indexed keywords

APPROXIMATION THEORY; HARMONIC ANALYSIS; SET THEORY;

HARMONIC APPROXIMATION; PARABOLIC SYSTEMS; PARTIAL REGULARITY; SINGULAR SETS;

NONLINEAR SYSTEMS;

EID: 25644433664     PISSN: 02941449     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.anihpc.2004.10.011     Document Type: Article

References (47)

1. E. Acerbi, G. Mingione, and G.A. Seregin Regularity results for parabolic systems related to a class of non-Newtonian fluids Ann. Inst. H. Poincaré Anal. Non Linéaire 21 2004 25 60
2. A.A. Arkhipova On a partial regularity up to the boundary of weak solutions to quasilinear parabolic systems with quadratic growth J. Math. Sci. (New York) 101 2000 3385 3397 (1997 in Russian)
3. n Ann. Acad. Sci. Fenn. Ser. A I 8 1983 257 324
4. 2,θ (Ω,δ) Ann. Mat. Pura Appl. 73 4 1996 55 102
5. S. Campanato On the nonlinear parabolic systems in divergence form. Hölder continuity and partial Hölder continuity of the solutions Ann. Mat. Pura Appl. 137 4 1984 83 122
6. p,θ) (Ω,δ) e loro proprietà Ann. Mat. Pura Appl. 69 4 1965 383 392
7. E. De Giorgi Frontiere orientate di misura minima Sem. Scuola Normale Superiore Pisa 1960-1961
8. E. De Giorgi Un esempio di estremali discontinue per un problema variazionale di tipo ellittico Boll. Un. Mat. Ital. 1 4 1968 135 137
9. E. DiBenedetto Degenerate Parabolic Equations Universitext vol. XV 1993 Springer-Verlag New York, NY p. 387
10. F. Duzaar, A. Gastel, and G. Mingione Elliptic systems, singular sets and Dini continuity Comm. Partial Differential Equations 29 2004
11. F. Duzaar, and J.F. Grotowski Optimal interior partial regularity for nonlinear elliptic systems: the method of A-harmonic approximation Manuscripta Math. 103 2000 267 298
12. F. Duzaar, and G. Mingione The p-harmonic approximation and the regularity of p-harmonic maps Calc. Var. Partial Differential Equations 20 2004 235 256
13. F. Duzaar, and G. Mingione Regularity for degenerate elliptic problems via p-harmonic approximation Ann. Inst. H. Poincaré Anal. Non Linéaire 21 2004
14. F. Duzaar, G. Mingione, K. Steffen, Second order parabolic systems with p-growth and regularity, in press
15. F. Duzaar, and K. Steffen Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals J. Reine Angew. Math. 546 2002 73 138
16. M. Giaquinta Multiple Integrals in the Calculus of Variations and Non Linear Elliptic Systems Ann. of Math. Stud. vol. 105 1983 Princeton University Press
17. M. Giaquinta, and M. Struwe On the partial regularity of weak solutions of nonlinear parabolic systems Math. Z. 179 1982 437 451
18. M. Giaquinta, and M. Struwe An optimal regularity result for a class of quasilinear parabolic systems Manuscripta Math. 36 1981 223 239
19. E. Giusti Direct Methods in the Calculus of Variations 2003 World Scientific Singapore
20. W. Hao, S. Leonardi, and J. Necas An example of irregular solution to a nonlinear Euler-Lagrange elliptic system with real analytic coefficients Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23 4 1996 57 67
21. O. John, and J. Stará Some (new) counterexamples of parabolic systems Comment. Math. Univ. Carolin. 36 1995 503 510
22. O. John, and J. Stará On the regularity of weak solutions to parabolic systems in two spatial dimensions Comm. Partial Differential Equations 23 1998 1159 1170
23. O. John, and J. Stará On the existence of time derivative of weak solutions to parabolic systems R. Salvi Navier-Stokes Equations. Theory and Numerical Methods, Proceedings of the International Conference Varenna, 1997 Pitman Res. Notes Math. Ser. vol. 388 1998 Longman Harlow 193 200
24. J. Kinnunen, and J.L. Lewis Higher integrability for parabolic systems of p-Laplacian type Duke Math. J. 102 2000 253 271
25. A. Koshelev Regularity Problem for Quasilinear Elliptic and Parabolic Systems Lecture Notes in Math. vol. 1614 1995 Springer
26. M. Kronz Partial regularity results for minimizers of quasiconvex functionals of higher order Ann. Inst. H. Poincaré Anal. Non Linéaire 19 2002 81 112
27. O.A. Ladyzhenskaya, V.A. Solonnikov, and N.N. Ural'tseva Linear and quasi-linear equations of parabolic type Transl. Math. Monographs vol. 23 1968 Amer. Math. Soc.
28. J.L. Lions Quelques methodes des resolution des problemes aux limites non lineaires 1969 Gauthier-Villars Paris
29. J. Malý, J. Stará, and O. John A note on the regularity of autonomous quasilinear elliptic and parabolic systems Comm. Partial Differential Equations 13 1988 895 903
30. G. Mingione The singular set of solutions to non-differentiable elliptic systems Arch. Ration. Mech. Anal. 166 2003 287 301
31. G. Mingione Bounds for the singular set of solutions to non linear elliptic systems Calc. Var. Partial Differential Equations 18 2003 373 400
32. M. Misawa Partial regularity results for evolutional p-Laplacian systems with natural growth Manuscripta Math. 109 2002 419 454
33. M. Misawa Local Hölder regularity of gradients for evolutional p-Laplacian systems Ann. Mat. Pura Appl. 181 4 2002 389 405
34. R. Moser Regularity for the approximated harmonic map equation and application to the heat flow for harmonic maps Math. Z. 243 2003 263 289
35. J. Naumann, J. Wolf, Interior integral estimates on weak solutions of nonlinear parabolic systems, Preprint 94-11, Inst. für Math., Humboldt University, 1994
36. J. Naumann, J. Wolf, and M. Wolff On the Hölder continuity of weak solutions to nonlinear parabolic systems in two space dimensions Comment. Math. Univ. Carolin. 39 1998 237 255
37. J. Necas, and V. Sverák On regularity of solutions of nonlinear parabolic systems Ann. Scuola Norm. Super. Pisa Cl. Sci. 18 4 1991 1 11
38. G.A. Seregin Interior regularity for solutions to the modified Navier-Stokes equations J. Math. Fluid Mech. 1 1999 235 281
39. p (0,T;B) Ann. Mat. Pura Appl. 146 4 1987 65 96
40. L. Simon Lectures on Geometric Measure Theory Proc. Centre Math. Anal. 1983 Austr. Nat. Univ. Canberra
41. L. Simon Theorems on Regularity and Singularity of Energy Minimizing Maps Lectures in Math. 1996 ETH Zürich, Birkhäuser Basel
42. J. Stará, O. John, and J. Malý Counterexample to the regularity of weak solution of the quasilinear parabolic system Comment. Math. Univ. Carolin. 27 1986 123 136
43. E.W. Stredulinsky Higher integrability from reverse Hölder inequalities Indiana Univ. Math. J. 29 1980 407 413
44. M. Struwe On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems Manuscripta Math. 35 1981 125 145
45. M. Struwe A counterexample in regularity theory for parabolic systems Czech. Math. J. 34 1984 183 188
46. V. Sverák, and X. Yan Non Lipschitz minimizers of smooth uniformly convex variational integrals Proc. Nat. Acad. Sci. USA 99 24 2002 15268 15276
47. 1,α-partial regularity of nonlinear parabolic systems J. Partial Differential Equations 5 1992 23 34