The boundary regularity of non-linear parabolic systems II
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 1, pp. 145-200.

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},
     pages = {145--200},
     publisher = {Elsevier},
     volume = {27},
     number = {1},
     year = {2010},
     doi = {10.1016/j.anihpc.2009.09.002},
     zbl = {1194.35085},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2009.09.002/}
}
TY  - JOUR
AU  - Bögelein, Verena
AU  - Duzaar, Frank
AU  - Mingione, Giuseppe
TI  - The boundary regularity of non-linear parabolic systems II
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2010
SP  - 145
EP  - 200
VL  - 27
IS  - 1
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2009.09.002/
DO  - 10.1016/j.anihpc.2009.09.002
LA  - en
ID  - AIHPC_2010__27_1_145_0
ER  - 
%0 Journal Article
%A Bögelein, Verena
%A Duzaar, Frank
%A Mingione, Giuseppe
%T The boundary regularity of non-linear parabolic systems II
%J Annales de l'I.H.P. Analyse non linéaire
%D 2010
%P 145-200
%V 27
%N 1
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2009.09.002/
%R 10.1016/j.anihpc.2009.09.002
%G en
%F 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) no. 1, pp. 145-200. doi : 10.1016/j.anihpc.2009.09.002. http://archive.numdam.org/articles/10.1016/j.anihpc.2009.09.002/

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

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

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

[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

[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

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

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

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

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

[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

[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

[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

[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 | EuDML | Numdam | Zbl

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

[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

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

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

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

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

[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 | Zbl

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

[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

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

[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

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

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

[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 | EuDML | Zbl

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

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

Cité par Sources :