no. 1
The boundary regularity of non-linear parabolic systems I
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 1, pp. 201-255

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.

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     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},
     pages = {201--255},
     year = {2010},
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     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2009.09.003/}
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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) no. 1, pp. 201-255. doi: 10.1016/j.anihpc.2009.09.003

[1] E. Acerbi, G. Mingione, Gradient estimates for a class of parabolic systems, Duke Math. J. 136 (2007), 285-320 | MR | Zbl

[2] E. Acerbi, G. Mingione, 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 | MR | EuDML | Zbl | Numdam

[3] A.A. Arkhipova, 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 | Zbl

[4] L. Beck, Partial regularity for weak solutions of nonlinear elliptic systems: The subquadratic case, Manuscripta Math. 123 no. 4 (2007), 453-491 | MR | Zbl

[5] V. Bögelein, Partial regularity and singular sets of solutions of higher order parabolic systems, Ann. Mat. Pura Appl. (4) 188 (2009), 61-122 | MR | Zbl

[6] V. Bögelein, F. Duzaar, G. Mingione, The boundary regularity of non-linear parabolic systems II, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 no. 1 (2010), 145-200 | MR | Zbl | Numdam

[7] S. Campanato, Equazioni paraboliche del secondo ordine e spazi 𝔏 2,θ (Ω,δ), Ann. Mat. Pura Appl. (4) 73 (1966), 55-102 | MR | Zbl

[8] S. Campanato, 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 | Zbl

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

[10] E. De Giorgi, Un esempio di estremali discontinue per un problema variazionale di tipo ellitico, Boll. Unione Mat. Ital. 4 (1968), 135-137 | MR | Zbl

[11] E. Dibenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York (1993) | MR | Zbl

[12] Y.Z. Chen, E. Dibenedetto, Boundary estimates for solutions of nonlinear degenerate parabolic systems, J. Reine Angew. Math. 395 (1989), 102-131 | MR | EuDML | Zbl

[13] F. Duzaar, A. Gastel, G. Mingione, Elliptic systems, singular sets and Dini continuity, Comm. Partial Differential Equations 29 (2004), 1215-1240 | MR | Zbl

[14] F. Duzaar, J.F. Grotowski, Optimal interior partial regularity for nonlinear elliptic systems: The method of a-harmonic approximation, Manuscripta Math. 103 (2000), 267-298 | MR | Zbl

[15] F. Duzaar, J.F. Grotowski, M. Kronz, Partial and full boundary regularity for minimizers of functionals with nonquadratic growth, J. Convex Anal. 11 (2004), 437-476 | MR | Zbl

[16] F. Duzaar, J. Kristensen, G. Mingione, The existence of regular boundary points for non-linear elliptic systems, J. Reine Angew. Math. 602 (2007), 17-58 | MR | Zbl

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

[18] F. Duzaar, G. Mingione, Second order parabolic systems, optimal regularity, and singular sets of solutions, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 705-751 | MR | EuDML | Zbl | Numdam

[19] F. Duzaar, G. Mingione, Harmonic type approximation lemmas, J. Math. Anal. Appl. 352 (2009), 301-335 | MR | Zbl

[20] F. Duzaar, G. Mingione, K. Steffen, Parabolic systems with polynomial growth and regularity, Mem. Amer. Math. Soc., in press | MR

[21] F.G. Duzaar, K. Steffen, Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals, J. Reine Angew. Math. 546 (2002) | MR | Zbl

[22] M. Giaquinta, A counter-example to the boundary regularity of solutions to quasilinear systems, Manuscripta Math. 24 (1978), 217-220 | MR | EuDML | Zbl

[23] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton Univ. Press, Princeton, NJ (1983) | MR | Zbl

[24] M. Giaquinta, G. Modica, Local existence for quasilinear parabolic systems under nonlinear boundary conditions, Ann. Mat. Pura Appl. (4) 149 (1987), 41-59 | MR | Zbl

[25] J. Grotowski, Boundary regularity results for non-linear elliptic systems in divergence form, Habilitationsschrift, 2000

[26] J. Grotowski, Boundary regularity for nonlinear elliptic systems, Calc. Var. Partial Differential Equations 15 (2002), 353-388 | MR | Zbl

[27] J. Kristensen, G. Mingione, The singular set of minima of integral functionals, Arch. Ration. Mech. Anal. 180 (2006), 331-398 | MR | Zbl

[28] J. Kristensen, G. Mingione, Boundary regularity in variational problems, in press | MR

[29] J. Kristensen, G. Mingione, Boundary regularity of minima, Rend. Lincei Mat. Appl. 19 (2008), 265-277 | MR | Zbl

[30] M. Kronz, Quasimonotone systems of higher order, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 6 (2003), 459-480 | MR | EuDML | Zbl

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

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

[33] G. Mingione, Regularity of minima: An invitation to the dark side of the calculus of variations, Appl. Math. 51 (2006), 355-425 | MR | EuDML | Zbl

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

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

[36] J. Stará, O. John, J. Malý, Counterexamples to the regularity of weak solutions of the quasilinear parabolic system, Comment. Math. Univ. Carolin. 27 (1986), 123-136 | MR | EuDML | Zbl

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