The boundary regularity of non-linear parabolic systems II
Bögelein, Verena ; Duzaar, Frank ; Mingione, Giuseppe
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010), p. 145-200 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

This is the second 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 establish higher fractional differentiability of solutions up to the boundary. Based on the necessary and sufficient condition for regular boundary points from the first part of Bögelein et al. (in this issue) [7] we achieve dimension estimates for the boundary singular set and eventually the almost everywhere regularity of solutions at the boundary.

@article{AIHPC_2010__27_1_145_0,
     author = {B\"ogelein, Verena and Duzaar, Frank and Mingione, Giuseppe},
     title = {The boundary regularity of non-linear parabolic systems II},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {27},
     year = {2010},
     pages = {145-200},
     doi = {10.1016/j.anihpc.2009.09.002},
     zbl = {1194.35085},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2010__27_1_145_0}
}

Bögelein, Verena; Duzaar, Frank; Mingione, Giuseppe. The boundary regularity of non-linear parabolic systems II. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) pp. 145-200. doi : 10.1016/j.anihpc.2009.09.002. http://gdmltest.u-ga.fr/item/AIHPC_2010__27_1_145_0/

[1] E. Acerbi, G. Mingione, Gradient estimates for a class of parabolic systems, Duke Math. J. 136 (2007), 285-320 | MR 2286632 | Zbl 1113.35105

[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 no. 1 (2004), 25-60 | Numdam | Zbl 1052.76004

[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 1698511 | Zbl 0969.35032

[4] L. Beck, 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] C. Bennett, R. Sharpley, Interpolation of Operators, Academic Press, Boston (1988) | MR 928802 | Zbl 0647.46057

[6] V. Bögelein, Partial regularity and singular sets of solutions of higher order parabolic systems, Ann. Mat. Pura Appl. 188 (2009), 61-122 | Zbl 1183.35158

[7] V. Bögelein, F. Duzaar, G. Mingione, The boundary regularity of non-linear parabolic systems I, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 no. 1 (2010), 201-255 | Numdam | Zbl 1194.35086

[8] V. Bögelein, M. Parviainen, Self-improving property of nonlinear higher order parabolic systems near the boundary, NoDEA Nonlinear Differential Equations Appl., doi:10.1007/s00030-009-0038-5 | MR 2596493

[9] B. Bojarski, T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in ℝ n , Ann. Acad. Sci. Fenn. Ser. A I 8 (1983), 257-324 | MR 731786 | Zbl 0548.30016

[10] E. Dibenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York (1993) | Zbl 0794.35090

[11] Y.Z. Chen, E. Dibenedetto, Boundary estimates for solutions of nonlinear degenerate parabolic systems, J. Reine Angew. Math. 395 (1989), 102-131 | Zbl 0661.35052

[12] A. Domokos, Differentiability of solutions for the non-degenerate p-Laplacian in the Heisenberg group, J. Differential Equations 204 (2004), 439-470 | MR 2085543 | Zbl 1065.35103

[13] 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 | Zbl 0971.35025

[14] 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 | Zbl 1066.49022

[15] F. Duzaar, J. Kristensen, G. Mingione, The existence of regular boundary points for non-linear elliptic systems, J. Reine Angew. Math. (Crelles J.) 602 (2007), 17-58 | Zbl 1214.35021

[16] 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 | Numdam | Zbl 1099.35042

[17] F. Duzaar, G. Mingione, Harmonic type approximation lemmas, J. Math. Anal. Appl. 352 (2009), 301-335 | Zbl 1172.35002

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

[19] F.G. Duzaar, K. Steffen, Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals, J. Reine Angew. Math. 546 (2002) | Zbl 0999.49024

[20] C. Fefferman, E.M. Stein, H p spaces of several variables, Acta Math. 129 (1972), 137-193 | MR 447953 | Zbl 0257.46078

[21] M. Giaquinta, A counter-example to the boundary regularity of solutions to quasilinear systems, Manuscripta Math. 24 (1978), 217-220 | Zbl 0373.35027

[22] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton Univ. Press, Princeton, NJ (1983) | Zbl 0516.49003

[23] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Company, Singapore (2003) | MR 1962933 | Zbl 1028.49001

[24] T. Iwaniec, On L p -integrability in PDE's and quasiregular mappings for large exponents, Ann. Acad. Sci. Fenn. Ser. A I 7 no. 2 (1982), 301-322 | MR 686647 | Zbl 0505.30011

[25] J. Kristensen, G. Mingione, The singular set of minima of integral functionals, Arch. Ration. Mech. Anal. 180 (2006), 331-398 | Zbl 1116.49010

[26] J. Kristensen, G. Mingione, Boundary regularity in variational problems, in press

[27] J. Kristensen, G. Mingione, Boundary regularity of minima, Rend. Lincei Mat. Appl. 19 (2008), 265-277 | Zbl 1194.49048

[28] G. Mingione, The singular set of solutions to non-differentiable elliptic systems, Arch. Ration. Mech. Anal. 166 (2003), 287-301 | Zbl 1142.35391

[29] G. Mingione, Bounds for the singular set of solutions to non linear elliptic systems, Calc. Var. Partial Differential Equations 18 (2003), 373-400 | Zbl 1045.35024

[30] G. Mingione, Regularity of minima: An invitation to the dark side of the calculus of variations, Appl. Math. 51 (2006), 355-425 | Zbl 1164.49324

[31] M. Parviainen, Global gradient estimates for degenerate parabolic equations in nonsmooth domains, Ann. Mat. Pura Appl. 188 no. 2 (2009), 333-358 | MR 2491806 | Zbl 1179.35080

[32] 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 | Zbl 0625.35047

[33] E.W. Stredulinsky, Higher integrability from reverse Hölder inequalities, Indiana Univ. Math. J. 29 (1980), 407-413 | MR 570689 | Zbl 0442.35064

[34] A. Zygmund, Trigonometric Series I, Cambridge Univ. Press, Cambridge (1977) | Zbl 0367.42001