Second order parabolic systems, optimal regularity, and singular sets of solutions

 Access to full text

 Full (PDF)

1. [1] Acerbi E., Mingione G., Seregin G.A., Regularity results for parabolic systems related to a class of non-Newtonian fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire21 (2004) 25-60. Zbl1052.76004MR2037246
2. [2] Arkhipova A.A., 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). Zbl0969.35032MR1698511
3. [3] Bojarski B., Iwaniec T., Analytical foundations of the theory of quasiconformal mappings in $R^n$, Ann. Acad. Sci. Fenn. Ser. A I8 (1983) 257-324. Zbl0548.30016MR731786
4. [4] Campanato S., Equazioni paraboliche del secondo ordine e spazi $L^{2,\theta }(\Omega ,\delta )$, Ann. Mat. Pura Appl.73 (4) (1996) 55-102. Zbl0144.14101MR213737
5. [5] Campanato S., 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. Zbl0704.35024MR772253
6. [6] Da Prato G., Spazi $L^{(p,\vartheta )}(\Omega ,\delta )$ e loro proprietà, Ann. Mat. Pura Appl.69 (4) (1965) 383-392. Zbl0145.16207MR192330
7. [7] De Giorgi E., Frontiere orientate di misura minima, Sem. Scuola Normale Superiore Pisa (1960–1961). Zbl0296.49031
8. [8] De Giorgi E., Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll. Un. Mat. Ital.1 (4) (1968) 135-137. Zbl0155.17603MR227827
9. [9] DiBenedetto E., Degenerate Parabolic Equations, Universitext, vol. XV, Springer-Verlag, New York, NY, 1993, p. 387. Zbl0794.35090MR1230384
10. [10] Duzaar F., Gastel A., Mingione G., Elliptic systems, singular sets and Dini continuity, Comm. Partial Differential Equations29 (2004). Zbl1140.35415MR2097582
11. [11] Duzaar F., Grotowski J.F., Optimal interior partial regularity for nonlinear elliptic systems: the method of A-harmonic approximation, Manuscripta Math.103 (2000) 267-298. Zbl0971.35025MR1802484
12. [12] Duzaar F., Mingione G., The p-harmonic approximation and the regularity of p-harmonic maps, Calc. Var. Partial Differential Equations20 (2004) 235-256. Zbl1142.35433MR2062943
13. [13] Duzaar F., Mingione G., Regularity for degenerate elliptic problems via p-harmonic approximation, Ann. Inst. H. Poincaré Anal. Non Linéaire21 (2004). Zbl1112.35078MR2086757
14. [14] F. Duzaar, G. Mingione, K. Steffen, Second order parabolic systems with p-growth and regularity, in press. Zbl1238.35001
15. [15] Duzaar F., Steffen K., Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals, J. Reine Angew. Math.546 (2002) 73-138. Zbl0999.49024MR1900994
16. [16] Giaquinta M., Multiple Integrals in the Calculus of Variations and Non Linear Elliptic Systems, Ann. of Math. Stud., vol. 105, Princeton University Press, 1983. Zbl0516.49003MR717034
17. [17] Giaquinta M., Struwe M., On the partial regularity of weak solutions of nonlinear parabolic systems, Math. Z.179 (1982) 437-451. Zbl0469.35028MR652852
18. [18] Giaquinta M., Struwe M., An optimal regularity result for a class of quasilinear parabolic systems, Manuscripta Math.36 (1981) 223-239. Zbl0475.35026MR641975
19. [19] Giusti E., Direct Methods in the Calculus of Variations, World Scientific, Singapore, 2003. Zbl1028.49001MR1962933
20. [20] Hao W., Leonardi S., Necas J., 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. Zbl0864.35031MR1401417
21. [21] John O., Stará J., Some (new) counterexamples of parabolic systems, Comment. Math. Univ. Carolin.36 (1995) 503-510. Zbl0846.35024MR1364491
22. [22] John O., Stará J., On the regularity of weak solutions to parabolic systems in two spatial dimensions, Comm. Partial Differential Equations23 (1998) 1159-1170. Zbl0937.35020MR1642595
23. [23] John O., Stará J., On the existence of time derivative of weak solutions to parabolic systems, in: Salvi R. (Ed.), Navier–Stokes Equations. Theory and Numerical Methods, Proceedings of the International Conference, Varenna, 1997, Pitman Res. Notes Math. Ser., vol. 388, Longman, Harlow, 1998, pp. 193-200. Zbl0940.35048MR1773597
24. [24] Kinnunen J., Lewis J.L., Higher integrability for parabolic systems of p-Laplacian type, Duke Math. J.102 (2000) 253-271. Zbl0994.35036MR1749438
25. [25] Koshelev A., Regularity Problem for Quasilinear Elliptic and Parabolic Systems, Lecture Notes in Math., vol. 1614, Springer, 1995. Zbl0847.35023MR1442954
26. [26] Kronz M., Partial regularity results for minimizers of quasiconvex functionals of higher order, Ann. Inst. H. Poincaré Anal. Non Linéaire19 (2002) 81-112. Zbl1010.49023MR1902546
27. [27] Ladyzhenskaya O.A., Solonnikov V.A., Ural'tseva N.N., Linear and quasi-linear equations of parabolic type, Transl. Math. Monographs, vol. 23, Amer. Math. Soc., 1968. Zbl0174.15403
28. [28] Lions J.L., Quelques methodes des resolution des problemes aux limites non lineaires, Gauthier-Villars, Paris, 1969. Zbl0189.40603MR259693
29. [29] Malý J., Stará J., John O., A note on the regularity of autonomous quasilinear elliptic and parabolic systems, Comm. Partial Differential Equations13 (1988) 895-903. Zbl0685.35024MR940961
30. [30] Mingione G., The singular set of solutions to non-differentiable elliptic systems, Arch. Ration. Mech. Anal.166 (2003) 287-301. Zbl1142.35391MR1961442
31. [31] Mingione G., Bounds for the singular set of solutions to non linear elliptic systems, Calc. Var. Partial Differential Equations18 (2003) 373-400. Zbl1045.35024MR2020367
32. [32] Misawa M., Partial regularity results for evolutional p-Laplacian systems with natural growth, Manuscripta Math.109 (2002) 419-454. Zbl1026.35025MR1946712
33. [33] Misawa M., Local Hölder regularity of gradients for evolutional p-Laplacian systems, Ann. Mat. Pura Appl.181 (4) (2002) 389-405. Zbl1223.35194MR1939688
34. [34] Moser R., Regularity for the approximated harmonic map equation and application to the heat flow for harmonic maps, Math. Z.243 (2003) 263-289. Zbl1027.58014MR1961867
35. [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. [36] Naumann J., Wolf J., Wolff M., On the Hölder continuity of weak solutions to nonlinear parabolic systems in two space dimensions, Comment. Math. Univ. Carolin.39 (1998) 237-255. Zbl0940.35046MR1651938
37. [37] Necas J., Sverák V., On regularity of solutions of nonlinear parabolic systems, Ann. Scuola Norm. Super. Pisa Cl. Sci.18 (4) (1991) 1-11. Zbl0735.35035MR1118218
38. [38] Seregin G.A., Interior regularity for solutions to the modified Navier–Stokes equations, J. Math. Fluid Mech.1 (1999) 235-281. Zbl0961.35106MR1738752
39. [39] Simon J., Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl.146 (4) (1987) 65-96. Zbl0629.46031MR916688
40. [40] Simon L., Lectures on Geometric Measure Theory, Proc. Centre Math. Anal., Austr. Nat. Univ., Canberra, 1983. Zbl0546.49019MR756417
41. [41] Simon L., Theorems on Regularity and Singularity of Energy Minimizing Maps, Lectures in Math., ETH Zürich, Birkhäuser, Basel, 1996. Zbl0864.58015MR1399562
42. [42] Stará J., John O., Malý J., Counterexample to the regularity of weak solution of the quasilinear parabolic system, Comment. Math. Univ. Carolin.27 (1986) 123-136. Zbl0625.35047MR843425
43. [43] Stredulinsky E.W., Higher integrability from reverse Hölder inequalities, Indiana Univ. Math. J.29 (1980) 407-413. Zbl0442.35064MR570689
44. [44] Struwe M., On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems, Manuscripta Math.35 (1981) 125-145. Zbl0519.35007MR627929
45. [45] Struwe M., A counterexample in regularity theory for parabolic systems, Czech. Math. J.34 (1984) 183-188. Zbl0573.35053MR743484
46. [46] Sverák V., Yan X., Non Lipschitz minimizers of smooth uniformly convex variational integrals, Proc. Nat. Acad. Sci. USA99 (24) (2002) 15268-15276. Zbl1106.49046MR1946762
47. [47] Zhong T., $C^{1,\alpha }$-partial regularity of nonlinear parabolic systems, J. Partial Differential Equations5 (1992) 23-34. Zbl0772.35009MR1155325

1. Arina A. Arkhipova, Jana Stará, Regularity problem for one class of nonlinear parabolic systems with non-smooth in time principal matrices
2. Arina A. Arkhipova, Jana Stará, A priori estimates for quasilinear parabolic systems with quadratic nonlinearities in the gradient
3. Giuseppe Mingione, Regularity of minima: an invitation to the Dark Side of the Calculus of Variations