Second order parabolic systems, optimal regularity, and singular sets of solutions
Annales de l'I.H.P. Analyse non linéaire, Volume 22 (2005) no. 6, pp. 705-751.
@article{AIHPC_2005__22_6_705_0,
     author = {Duzaar, Frank and Mingione, Giuseppe},
     title = {Second order parabolic systems, optimal regularity, and singular sets of solutions},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {705--751},
     publisher = {Elsevier},
     volume = {22},
     number = {6},
     year = {2005},
     doi = {10.1016/j.anihpc.2004.10.011},
     zbl = {1099.35042},
     mrnumber = {2172857},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2004.10.011/}
}
TY  - JOUR
AU  - Duzaar, Frank
AU  - Mingione, Giuseppe
TI  - Second order parabolic systems, optimal regularity, and singular sets of solutions
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2005
DA  - 2005///
SP  - 705
EP  - 751
VL  - 22
IS  - 6
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2004.10.011/
UR  - https://zbmath.org/?q=an%3A1099.35042
UR  - https://www.ams.org/mathscinet-getitem?mr=2172857
UR  - https://doi.org/10.1016/j.anihpc.2004.10.011
DO  - 10.1016/j.anihpc.2004.10.011
LA  - en
ID  - AIHPC_2005__22_6_705_0
ER  - 
%0 Journal Article
%A Duzaar, Frank
%A Mingione, Giuseppe
%T Second order parabolic systems, optimal regularity, and singular sets of solutions
%J Annales de l'I.H.P. Analyse non linéaire
%D 2005
%P 705-751
%V 22
%N 6
%I Elsevier
%U https://doi.org/10.1016/j.anihpc.2004.10.011
%R 10.1016/j.anihpc.2004.10.011
%G en
%F AIHPC_2005__22_6_705_0
Duzaar, Frank; Mingione, Giuseppe. Second order parabolic systems, optimal regularity, and singular sets of solutions. Annales de l'I.H.P. Analyse non linéaire, Volume 22 (2005) no. 6, pp. 705-751. doi : 10.1016/j.anihpc.2004.10.011. http://www.numdam.org/articles/10.1016/j.anihpc.2004.10.011/

[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 Linéaire 21 (2004) 25-60. | Numdam | MR | Zbl

[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). | MR | Zbl

[3] Bojarski B., Iwaniec T., Analytical foundations of the theory of quasiconformal mappings in R n , Ann. Acad. Sci. Fenn. Ser. A I 8 (1983) 257-324. | MR | Zbl

[4] Campanato S., Equazioni paraboliche del secondo ordine e spazi L 2,θ (Ω,δ), Ann. Mat. Pura Appl. 73 (4) (1996) 55-102. | MR | Zbl

[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. | MR | Zbl

[6] Da Prato G., Spazi L (p,ϑ) (Ω,δ) e loro proprietà, Ann. Mat. Pura Appl. 69 (4) (1965) 383-392. | MR | Zbl

[7] De Giorgi E., Frontiere orientate di misura minima, Sem. Scuola Normale Superiore Pisa (1960-1961). | Zbl

[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. | MR | Zbl

[9] Dibenedetto E., Degenerate Parabolic Equations, Universitext, vol. XV, Springer-Verlag, New York, NY, 1993, p. 387. | MR | Zbl

[10] Duzaar F., Gastel A., Mingione G., Elliptic systems, singular sets and Dini continuity, Comm. Partial Differential Equations 29 (2004). | MR | Zbl

[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. | MR | Zbl

[12] Duzaar F., Mingione G., The p-harmonic approximation and the regularity of p-harmonic maps, Calc. Var. Partial Differential Equations 20 (2004) 235-256. | MR | Zbl

[13] Duzaar F., Mingione G., Regularity for degenerate elliptic problems via p-harmonic approximation, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004). | Numdam | MR | Zbl

[14] F. Duzaar, G. Mingione, K. Steffen, Second order parabolic systems with p-growth and regularity, in press.

[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. | MR | Zbl

[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. | MR | Zbl

[17] Giaquinta M., Struwe M., On the partial regularity of weak solutions of nonlinear parabolic systems, Math. Z. 179 (1982) 437-451. | MR | Zbl

[18] Giaquinta M., Struwe M., An optimal regularity result for a class of quasilinear parabolic systems, Manuscripta Math. 36 (1981) 223-239. | MR | Zbl

[19] Giusti E., Direct Methods in the Calculus of Variations, World Scientific, Singapore, 2003. | MR | Zbl

[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. | Numdam | MR | Zbl

[21] John O., Stará J., Some (new) counterexamples of parabolic systems, Comment. Math. Univ. Carolin. 36 (1995) 503-510. | MR | Zbl

[22] John O., Stará J., On the regularity of weak solutions to parabolic systems in two spatial dimensions, Comm. Partial Differential Equations 23 (1998) 1159-1170. | MR | Zbl

[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. | MR | Zbl

[24] Kinnunen J., Lewis J.L., Higher integrability for parabolic systems of p-Laplacian type, Duke Math. J. 102 (2000) 253-271. | MR | Zbl

[25] Koshelev A., Regularity Problem for Quasilinear Elliptic and Parabolic Systems, Lecture Notes in Math., vol. 1614, Springer, 1995. | MR | Zbl

[26] Kronz M., Partial regularity results for minimizers of quasiconvex functionals of higher order, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002) 81-112. | Numdam | MR | Zbl

[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. | Zbl

[28] Lions J.L., Quelques methodes des resolution des problemes aux limites non lineaires, Gauthier-Villars, Paris, 1969. | MR | Zbl

[29] Malý J., Stará J., John O., A note on the regularity of autonomous quasilinear elliptic and parabolic systems, Comm. Partial Differential Equations 13 (1988) 895-903. | MR | Zbl

[30] Mingione G., The singular set of solutions to non-differentiable elliptic systems, Arch. Ration. Mech. Anal. 166 (2003) 287-301. | MR | Zbl

[31] Mingione G., Bounds for the singular set of solutions to non linear elliptic systems, Calc. Var. Partial Differential Equations 18 (2003) 373-400. | MR | Zbl

[32] Misawa M., Partial regularity results for evolutional p-Laplacian systems with natural growth, Manuscripta Math. 109 (2002) 419-454. | MR | Zbl

[33] Misawa M., Local Hölder regularity of gradients for evolutional p-Laplacian systems, Ann. Mat. Pura Appl. 181 (4) (2002) 389-405. | MR | Zbl

[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. | MR | Zbl

[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] 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. | MR | Zbl

[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. | Numdam | MR | Zbl

[38] Seregin G.A., Interior regularity for solutions to the modified Navier-Stokes equations, J. Math. Fluid Mech. 1 (1999) 235-281. | MR | Zbl

[39] Simon J., Compact sets in the space L p (0,T;B), Ann. Mat. Pura Appl. 146 (4) (1987) 65-96. | MR | Zbl

[40] Simon L., Lectures on Geometric Measure Theory, Proc. Centre Math. Anal., Austr. Nat. Univ., Canberra, 1983. | MR | Zbl

[41] Simon L., Theorems on Regularity and Singularity of Energy Minimizing Maps, Lectures in Math., ETH Zürich, Birkhäuser, Basel, 1996. | MR | Zbl

[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. | MR | Zbl

[43] Stredulinsky E.W., Higher integrability from reverse Hölder inequalities, Indiana Univ. Math. J. 29 (1980) 407-413. | MR | Zbl

[44] Struwe M., On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems, Manuscripta Math. 35 (1981) 125-145. | MR | Zbl

[45] Struwe M., A counterexample in regularity theory for parabolic systems, Czech. Math. J. 34 (1984) 183-188. | MR | Zbl

[46] Sverák V., Yan X., Non Lipschitz minimizers of smooth uniformly convex variational integrals, Proc. Nat. Acad. Sci. USA 99 (24) (2002) 15268-15276. | MR | Zbl

[47] Zhong T., C 1,α -partial regularity of nonlinear parabolic systems, J. Partial Differential Equations 5 (1992) 23-34. | MR | Zbl

Cited by Sources: