Regularity problem for one class of nonlinear parabolic systems with non-smooth in time principal matrices

Arina A. Arkhipova; Jana Stará

Commentationes Mathematicae Universitatis Carolinae (2019)

  • Volume: 60, Issue: 2, page 231-267
  • ISSN: 0010-2628

Abstract

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Partial regularity of solutions to a class of second order nonlinear parabolic systems with non-smooth in time principal matrices is proved in the paper. The coefficients are assumed to be measurable and bounded in the time variable and VMO-smooth in the space variables uniformly with respect to time. To prove the result, we apply the so-called A ( t ) -caloric approximation method. The method was applied by the authors earlier to study regularity of quasilinear systems.

How to cite

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Arkhipova, Arina A., and Stará, Jana. "Regularity problem for one class of nonlinear parabolic systems with non-smooth in time principal matrices." Commentationes Mathematicae Universitatis Carolinae 60.2 (2019): 231-267. <http://eudml.org/doc/294240>.

@article{Arkhipova2019,
abstract = {Partial regularity of solutions to a class of second order nonlinear parabolic systems with non-smooth in time principal matrices is proved in the paper. The coefficients are assumed to be measurable and bounded in the time variable and VMO-smooth in the space variables uniformly with respect to time. To prove the result, we apply the so-called $A(t)$-caloric approximation method. The method was applied by the authors earlier to study regularity of quasilinear systems.},
author = {Arkhipova, Arina A., Stará, Jana},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {nonlinear parabolic systems; regularity problem},
language = {eng},
number = {2},
pages = {231-267},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Regularity problem for one class of nonlinear parabolic systems with non-smooth in time principal matrices},
url = {http://eudml.org/doc/294240},
volume = {60},
year = {2019},
}

TY - JOUR
AU - Arkhipova, Arina A.
AU - Stará, Jana
TI - Regularity problem for one class of nonlinear parabolic systems with non-smooth in time principal matrices
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2019
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 60
IS - 2
SP - 231
EP - 267
AB - Partial regularity of solutions to a class of second order nonlinear parabolic systems with non-smooth in time principal matrices is proved in the paper. The coefficients are assumed to be measurable and bounded in the time variable and VMO-smooth in the space variables uniformly with respect to time. To prove the result, we apply the so-called $A(t)$-caloric approximation method. The method was applied by the authors earlier to study regularity of quasilinear systems.
LA - eng
KW - nonlinear parabolic systems; regularity problem
UR - http://eudml.org/doc/294240
ER -

References

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  1. Arkhipova A. A., On the regularity of the solutions of the Neumann problem for quasilinear parabolic systems, Izv. Ross. Akad. Nauk Ser. Mat. 58 (1994), no. 5, 3–25 (Russian); translation in Russian Acad. Sci. Izv. Math. 45 (1995), no. 2, 231–253. MR1307308
  2. Arkhipova A. A., Reverse Hölder inequalities with boundary integrals and L p -estimates for solutions of nonlinear elliptic and parabolic boundary value problems, Nonlinear Evolution Equations, Amer. Math. Soc. Transl. Ser. 2, 164, Adv. Math. Sci., 22, Amer. Math. Soc. (1995), 15–42. MR1334137
  3. Arkhipova A. A., 10.1007/s00229-016-0838-y, Manuscripta Math. 151 (2016), no. 3–4, 519–548. MR3556832DOI10.1007/s00229-016-0838-y
  4. Arkhipova A. A., 10.1134/S0965542517030034, Comput. Math. Math. Phys. 57 (2017), no. 3, 476–496. MR3636034DOI10.1134/S0965542517030034
  5. Arkhipova A. A., Stará J., Boundary partial regularity for solutions of quasilinear parabolic systems with non smooth in time principal matrix, Nonlinear Anal. 120 (2015), 236–261. MR3348057
  6. Arkhipova A. A., Stará J., 10.1515/forum-2015-0222, Forum Math. 29 (2017), no. 5, 1039–1064. MR3692026DOI10.1515/forum-2015-0222
  7. Arkhipova A. A., Stará J., Regularity problem for 2 m -order quasilinear parabolic systems with non smooth in time principal matrix. ( A ( t ) , m ) -caloric approximation method, Topol. Methods Nonlinear Anal. 52 (2018), no. 1, 111–146. MR3867982
  8. Arkhipova A. A., Stará J., John O., Partial regularity for solutions of quasilinear parabolic systems with nonsmooth in time principal matrix, Nonlinear Anal. 95 (2014), 421–435. MR3130534
  9. Bögelein V., Duzaar F., Mingione G., 10.1016/j.anihpc.2009.09.003, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), no. 1, 201–255. MR2580509DOI10.1016/j.anihpc.2009.09.003
  10. Campanato S., 10.1007/BF02415082, Ann. Mat. Pura Appl. (4) 73 (1966), 55–102 (Italian). MR0213737DOI10.1007/BF02415082
  11. Campanato S., 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. MR0772253
  12. Diening L., Lengeler D., Stroffolini B., Verde A., 10.1137/120870554, SIAM J. Math. Anal. 44 (2012), no. 5, 3594–3616. MR3023424DOI10.1137/120870554
  13. Diening L., Schwarzacher S., Stroffolini B., Verde A., 10.1007/s00526-017-1209-6, Calc. Var. Partial Differential Equations 56 (2017), no. 4, Art. 120, 27 pages. MR3672391DOI10.1007/s00526-017-1209-6
  14. Dong H., Kim D., Parabolic and elliptic systems with VMO coefficients, Methods Appl. Anal. 16 (2009), no. 3, 365–388. MR2650802
  15. Dong H., Kim D., 10.1080/03605302.2011.571746, Communic. Partial Differential Equations 36 (2011), no. 10, 1750–1777. MR2832162DOI10.1080/03605302.2011.571746
  16. Dong H., Kim D., 10.1007/s00205-010-0345-3, Arch. Ration. Mech. Anal. 199 (2011), no. 3, 889–941. MR2771670DOI10.1007/s00205-010-0345-3
  17. Duzaar F., Grotowski J. F., 10.1007/s002290070007, Manuscripta Math. 103 (2000), no. 3, 267–298. MR1802484DOI10.1007/s002290070007
  18. Duzaar F., Mingione G., 10.1016/j.anihpc.2004.10.011, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), no. 6, 705–751. MR2172857DOI10.1016/j.anihpc.2004.10.011
  19. Duzaar F., Steffen K., Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals, J. Reine Angew. Math. 546 (2002), 73–138. MR1900994
  20. Gehring F. W., 10.1007/BF02392268, Acta Math. 130 (1973), 265–277. MR0402038DOI10.1007/BF02392268
  21. Giaquinta M., Giusti E., 10.1007/BF02414914, Ann. Mat. Pura Appl. (4) 97 (1973), 253–266. MR0338568DOI10.1007/BF02414914
  22. Giaquinta M., Struwe M., 10.1007/BF01215058, Math. Z. 179 (1982), no. 4, 437–451. MR0652852DOI10.1007/BF01215058
  23. Kinnunen J., Lewis J. L., 10.1215/S0012-7094-00-10223-2, Duke Math. J. 102 (2000), no. 2, 253–271. MR1749438DOI10.1215/S0012-7094-00-10223-2
  24. Krylov N. V., 10.1080/03605300600781626, Comm. Partial Differential Equations 32 (2007), no. 1–3, 453–475. MR2304157DOI10.1080/03605300600781626
  25. Krylov N. V., 10.1090/gsm/096, Graduate Studies in Mathematics, 96, American Mathematical Society, Providence, 2008. MR2435520DOI10.1090/gsm/096
  26. Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., 10.1090/mmono/023, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, 1968. MR0241822DOI10.1090/mmono/023
  27. Mingione G., 10.1007/s00205-002-0231-8, Arch. Ration. Mech. Anal. 166 (2003), no. 4, 287–301. MR1961442DOI10.1007/s00205-002-0231-8

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