This is the first part of a work aimed at establishing that for solutions to Cauchy–Dirichlet problems involving general non-linear systems of parabolic type, almost every parabolic boundary point is a Hölder continuity point for the spatial gradient of solutions. Here we develop the basic necessary and sufficient condition for establishing the regular nature of a boundary point.
@article{AIHPC_2010__27_1_201_0, author = {B\"ogelein, Verena and Duzaar, Frank and Mingione, Giuseppe}, title = {The boundary regularity of non-linear parabolic systems I}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, volume = {27}, year = {2010}, pages = {201-255}, doi = {10.1016/j.anihpc.2009.09.003}, mrnumber = {2580509}, zbl = {1194.35086}, language = {en}, url = {http://dml.mathdoc.fr/item/AIHPC_2010__27_1_201_0} }
Bögelein, Verena; Duzaar, Frank; Mingione, Giuseppe. The boundary regularity of non-linear parabolic systems I. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) pp. 201-255. doi : 10.1016/j.anihpc.2009.09.003. http://gdmltest.u-ga.fr/item/AIHPC_2010__27_1_201_0/
[1] Gradient estimates for a class of parabolic systems, Duke Math. J. 136 (2007), 285-320 | MR 2286632 | Zbl 1113.35105
, ,[2] 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 2037246 | Zbl 1052.76004
, , ,[3] On a partial regularity up to the boundary of weak solutions to quasilinear parabolic systems with quadratic growth, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 249 no. 5 (1997), 20-39 | MR 1698511 | Zbl 0969.35032
,[4] Partial regularity for weak solutions of nonlinear elliptic systems: The subquadratic case, Manuscripta Math. 123 no. 4 (2007), 453-491 | MR 2320739 | Zbl 1151.35023
,[5] Partial regularity and singular sets of solutions of higher order parabolic systems, Ann. Mat. Pura Appl. (4) 188 (2009), 61-122 | MR 2447930 | Zbl 1183.35158
,[6] The boundary regularity of non-linear parabolic systems II, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 no. 1 (2010), 145-200 | Numdam | MR 2580508 | Zbl 1194.35085
, , ,[7] Equazioni paraboliche del secondo ordine e spazi , Ann. Mat. Pura Appl. (4) 73 (1966), 55-102 | MR 213737 | Zbl 0144.14101
,[8] On the nonlinear parabolic systems in divergence form. Hölder continuity and partial Hölder continuity of the solutions, Ann. Mat. Pura Appl. (4) 137 (1984), 83-122 | MR 772253 | Zbl 0704.35024
,[9] Frontiere orientate di misura minima, Sem. Scuola Normale Superiore, Pisa (1960)(1961) | MR 179651 | Zbl 0296.49031
,[10] Un esempio di estremali discontinue per un problema variazionale di tipo ellitico, Boll. Unione Mat. Ital. 4 (1968), 135-137 | MR 227827 | Zbl 0155.17603
,[11] Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York (1993) | MR 1230384 | Zbl 0794.35090
,[12] Boundary estimates for solutions of nonlinear degenerate parabolic systems, J. Reine Angew. Math. 395 (1989), 102-131 | MR 983061 | Zbl 0661.35052
, ,[13] Elliptic systems, singular sets and Dini continuity, Comm. Partial Differential Equations 29 (2004), 1215-1240 | MR 2097582 | Zbl 1140.35415
, , ,[14] Optimal interior partial regularity for nonlinear elliptic systems: The method of a-harmonic approximation, Manuscripta Math. 103 (2000), 267-298 | MR 1802484 | Zbl 0971.35025
, ,[15] Partial and full boundary regularity for minimizers of functionals with nonquadratic growth, J. Convex Anal. 11 (2004), 437-476 | MR 2158914 | Zbl 1066.49022
, , ,[16] The existence of regular boundary points for non-linear elliptic systems, J. Reine Angew. Math. 602 (2007), 17-58 | MR 2300451 | Zbl 1214.35021
, , ,[17] The p-harmonic approximation and the regularity of p-harmonic maps, Calc. Var. Partial Differential Equations 20 (2004), 235-256 | MR 2062943 | Zbl 1142.35433
, ,[18] Second order parabolic systems, optimal regularity, and singular sets of solutions, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 705-751 | Numdam | MR 2172857 | Zbl 1099.35042
, ,[19] Harmonic type approximation lemmas, J. Math. Anal. Appl. 352 (2009), 301-335 | MR 2499905 | Zbl 1172.35002
, ,[20] F. Duzaar, G. Mingione, K. Steffen, Parabolic systems with polynomial growth and regularity, Mem. Amer. Math. Soc., in press | MR 2866816
[21] Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals, J. Reine Angew. Math. 546 (2002) | MR 1900994 | Zbl 0999.49024
, ,[22] A counter-example to the boundary regularity of solutions to quasilinear systems, Manuscripta Math. 24 (1978), 217-220 | MR 492658 | Zbl 0373.35027
,[23] Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton Univ. Press, Princeton, NJ (1983) | MR 717034 | Zbl 0516.49003
,[24] Local existence for quasilinear parabolic systems under nonlinear boundary conditions, Ann. Mat. Pura Appl. (4) 149 (1987), 41-59 | MR 932775 | Zbl 0655.35049
, ,[25] J. Grotowski, Boundary regularity results for non-linear elliptic systems in divergence form, Habilitationsschrift, 2000
[26] Boundary regularity for nonlinear elliptic systems, Calc. Var. Partial Differential Equations 15 (2002), 353-388 | MR 1938819 | Zbl 1148.35315
,[27] The singular set of minima of integral functionals, Arch. Ration. Mech. Anal. 180 (2006), 331-398 | MR 2214961 | Zbl 1116.49010
, ,[28] J. Kristensen, G. Mingione, Boundary regularity in variational problems, in press | MR 2721587
[29] Boundary regularity of minima, Rend. Lincei Mat. Appl. 19 (2008), 265-277 | MR 2465679 | Zbl 1194.49048
, ,[30] Quasimonotone systems of higher order, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 6 (2003), 459-480 | MR 1988216 | Zbl 1150.35385
,[31] The singular set of solutions to non-differentiable elliptic systems, Arch. Ration. Mech. Anal. 166 (2003), 287-301 | MR 1961442 | Zbl 1142.35391
,[32] Bounds for the singular set of solutions to non linear elliptic systems, Calc. Var. Partial Differential Equations 18 (2003), 373-400 | MR 2020367 | Zbl 1045.35024
,[33] Regularity of minima: An invitation to the dark side of the calculus of variations, Appl. Math. 51 (2006), 355-425 | MR 2291779 | Zbl 1164.49324
,[34] Compact sets in the space , Ann. Mat. Pura Appl. (4) 146 (1987), 65-96 | MR 916688 | Zbl 0629.46031
,[35] Theorems on Regularity and Singularity of Energy Minimizing Maps, Lectures Math. ETH Zürich, Birkhäuser, Basel (1996) | MR 1399562
,[36] Counterexamples to the regularity of weak solutions of the quasilinear parabolic system, Comment. Math. Univ. Carolin. 27 (1986), 123-136 | MR 843425 | Zbl 0625.35047
, , ,