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On global \(L^q\) estimates for systems with p-growth in rough domains

  • Miroslav Bulíček
  • Sun-Sig Byun
  • Petr Kaplický
  • Jehan Oh
  • Sebastian SchwarzacherEmail author
Article
  • 104 Downloads

Abstract

We study regularity results for nonlinear parabolic systems of p-Laplacian type with inhomogeneous boundary and initial data, with \(p\in (\frac{2n}{n+2},\infty )\). We show bounds on the gradient of solutions in the Lebesgue-spaces with arbitrary large integrability exponents and natural dependences on the right hand side and the boundary data. In particular, we provide a new proof of the global non-linear Calderón–Zygmund theory for such systems. This extends the global result of Bögelein (Calc Var Partial Differ Equ 51(3–4):555–596, 2014) to very rough domains and more general boundary values. Our method makes use of direct estimates on the solution minus its boundary values and hence is considerably shorter than the available higher integrability results. Technically interesting is the fact that our parabolic estimates have no scaling deficit with respect to the leading order term. Moreover, in the singular case, \(p\in (\frac{2n}{n+2},2]\), any scaling deficit can be omitted.

Mathematics Subject Classification

35K51 35K55 35B65 35A01 

Notes

Acknowledgements

The authors would like to thank the referee for the very valuable suggestions and comments which led to improvement of the paper. M. Bulíček’s and S. Schwarzacher’s work is supported by the project LL1202 financed by the Ministry of Education, Youth and Sports, Czech Republic and by the University Centre for Mathematical Modelling, Applied Analysis and Computational Mathematics (Math MAC). M. Bulíček, P. Kaplickýand S. Schwarzacher are members of the Nečas Center for Mathematical Modeling. S. Byun was supported by NRF-2017R1A2B2003877.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Miroslav Bulíček
    • 1
  • Sun-Sig Byun
    • 2
  • Petr Kaplický
    • 3
  • Jehan Oh
    • 4
  • Sebastian Schwarzacher
    • 3
    • 5
    Email author
  1. 1.Mathematical Institute, Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic
  2. 2.Department of Mathematical Sciences and Research Institute of MathematicsSeoul National UniversitySeoulKorea
  3. 3.Department of Mathematical Analysis, Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic
  4. 4.Department of MathematicsKyungpook National UniversityDaeguKorea
  5. 5.Institute of Applied MathematicsUniversity of BonnBonnGermany

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