×
Bögelein, Verena; Duzaar, Frank; Mingione, Giuseppe

The boundary regularity of non-linear parabolic systems. II. (English) Zbl 1194.35085

Ann. Inst. Henri Poincaré, Anal. Non Linéaire 27, No. 1, 145-200 (2010).
This is a sequel of [V. Bögelein, F. Duzaar and G. Mingione, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 27, No. 1, 201–255 (2010; Zbl 1194.35086)], where the authors have proved a rather general boundary regularity criterion for nonlinear parabolic systems of the form \(u_t-\text{div}\,a(x,t,u,Du)=0\).
The boundary data are of Cauchy-Dirichlet type, given by a function \(g\) on the initial time slice and the lateral boundary. The function \(g\) is assumed to be continuous with Hölder continuous spatial derivative and time derivative in some appropriate Morrey space. The growth, regularity and ellipticity assumptions on \(a\) are somewhat stronger than in the first paper, but completely natural. The authors prove existence of regular initial and lateral boundary points for weak solutions, which has been open except for very special cases.
More precisely, they estimate the parabolic Hausdorff dimension of the singular set in terms of the Hölder exponents from the assumptions, finding that, once these exponents are not too small, the singular sets have lower dimension than the boundary components they are subsets of. Which means that almost every boundary point must be regular.
The proofs involve technically involved continuations of methods many of which have been developped by the authors in earlier work, including fractional difference quotients and the relations between fractional Sobolev and Nikolskii spaces.
Reviewer: Andreas Gastel (Erlangen)

Cited in 2 Reviews
Cited in 25 Documents

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35K51 Initial-boundary value problems for second-order parabolic systems
35K59 Quasilinear parabolic equations
35K65 Degenerate parabolic equations

Keywords:

parabolic system; Cauchy-Dirichlet problem; boundary regularity; Morrey space; parabolic Hausdorff dimension of the singular set; fractional difference quotients; fractional Sobolev and Nikolskii spaces

Citations:

Zbl 1194.35086
PDF BibTeX XML Cite
\textit{V. Bögelein} et al., Ann. Inst. Henri Poincaré, Anal. Non Linéaire 27, No. 1, 145--200 (2010; Zbl 1194.35085)
Full Text: DOI

References:

[1] Acerbi, E.; Mingione, G., Gradient estimates for a class of parabolic systems, Duke Math. J., 136, 285-320 (2007) · Zbl 1113.35105
[2] 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, 1, 25-60 (2004) · Zbl 1052.76004
[3] Arkhipova, A. A., 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, 5, 20-39 (1997) · Zbl 0969.35032
[4] Beck, L., Partial regularity for weak solutions of nonlinear elliptic systems: The subquadratic case, Manuscripta Math., 123, 4, 453-491 (2007) · Zbl 1151.35023
[5] Bennett, C.; Sharpley, R., Interpolation of Operators (1988), Academic Press: Academic Press Boston · Zbl 0647.46057
[6] Bögelein, V., Partial regularity and singular sets of solutions of higher order parabolic systems, Ann. Mat. Pura Appl., 188, 61-122 (2009) · Zbl 1183.35158
[7] Bögelein, V.; Duzaar, F.; Mingione, G., The boundary regularity of non-linear parabolic systems I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27, 1, 201-255 (2010), (in this issue) · Zbl 1194.35086
[9] Bojarski, B.; Iwaniec, T., Analytical foundations of the theory of quasiconformal mappings in \(R^n\), Ann. Acad. Sci. Fenn. Ser. A I, 8, 257-324 (1983) · Zbl 0548.30016
[10] DiBenedetto, E., Degenerate Parabolic Equations, Universitext (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0794.35090
[11] Chen, Y. Z.; DiBenedetto, E., Boundary estimates for solutions of nonlinear degenerate parabolic systems, J. Reine Angew. Math., 395, 102-131 (1989) · Zbl 0661.35052
[12] Domokos, A., Differentiability of solutions for the non-degenerate \(p\)-Laplacian in the Heisenberg group, J. Differential Equations, 204, 439-470 (2004) · Zbl 1065.35103
[13] Duzaar, F.; Grotowski, J. F., Optimal interior partial regularity for nonlinear elliptic systems: The method of \(a\)-harmonic approximation, Manuscripta Math., 103, 267-298 (2000) · Zbl 0971.35025
[14] Duzaar, F.; Grotowski, J. F.; Kronz, M., Partial and full boundary regularity for minimizers of functionals with nonquadratic growth, J. Convex Anal., 11, 437-476 (2004) · Zbl 1066.49022
[15] Duzaar, F.; Kristensen, J.; Mingione, G., The existence of regular boundary points for non-linear elliptic systems, J. Reine Angew. Math. (Crelles J.), 602, 17-58 (2007) · Zbl 1214.35021
[16] Duzaar, F.; Mingione, G., Second order parabolic systems, optimal regularity, and singular sets of solutions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22, 705-751 (2005) · Zbl 1099.35042
[17] Duzaar, F.; Mingione, G., Harmonic type approximation lemmas, J. Math. Anal. Appl., 352, 301-335 (2009) · Zbl 1172.35002
[19] Duzaar, F. G.; Steffen, K., Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals, J. Reine Angew. Math., 546 (2002) · Zbl 0999.49024
[20] Fefferman, C.; Stein, E. M., \(H^p\) spaces of several variables, Acta Math., 129, 137-193 (1972) · Zbl 0257.46078
[21] Giaquinta, M., A counter-example to the boundary regularity of solutions to quasilinear systems, Manuscripta Math., 24, 217-220 (1978) · Zbl 0373.35027
[22] Giaquinta, M., Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems (1983), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0516.49003
[23] Giusti, E., Direct Methods in the Calculus of Variations (2003), World Scientific Publishing Company: World Scientific Publishing Company Singapore · Zbl 1028.49001
[24] Iwaniec, T., On \(L^p\)-integrability in PDE’s and quasiregular mappings for large exponents, Ann. Acad. Sci. Fenn. Ser. A I, 7, 2, 301-322 (1982) · Zbl 0505.30011
[25] Kristensen, J.; Mingione, G., The singular set of minima of integral functionals, Arch. Ration. Mech. Anal., 180, 331-398 (2006) · Zbl 1116.49010
[27] Kristensen, J.; Mingione, G., Boundary regularity of minima, Rend. Lincei Mat. Appl., 19, 265-277 (2008) · Zbl 1194.49048
[28] Mingione, G., The singular set of solutions to non-differentiable elliptic systems, Arch. Ration. Mech. Anal., 166, 287-301 (2003) · Zbl 1142.35391
[29] Mingione, G., Bounds for the singular set of solutions to non linear elliptic systems, Calc. Var. Partial Differential Equations, 18, 373-400 (2003) · Zbl 1045.35024
[30] Mingione, G., Regularity of minima: An invitation to the dark side of the calculus of variations, Appl. Math., 51, 355-425 (2006) · Zbl 1164.49324
[31] Parviainen, M., Global gradient estimates for degenerate parabolic equations in nonsmooth domains, Ann. Mat. Pura Appl., 188, 2, 333-358 (2009) · Zbl 1179.35080
[32] Stará, J.; John, O.; Malý, J., Counterexamples to the regularity of weak solutions of the quasilinear parabolic system, Comment. Math. Univ. Carolin., 27, 123-136 (1986) · Zbl 0625.35047
[33] Stredulinsky, E. W., Higher integrability from reverse Hölder inequalities, Indiana Univ. Math. J., 29, 407-413 (1980) · Zbl 0442.35064
[34] Zygmund, A., Trigonometric Series I (1977), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0367.42001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.