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Sharp regularity estimates for quasilinear evolution equations

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Abstract

We establish sharp geometric C1+α regularity estimates for bounded weak solutions of evolution equations of p-Laplacian type. Our approach is based on geometric tangential methods, and makes use of a systematic oscillation mechanism combined with an adjusted intrinsic scaling argument.

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References

  1. E. Acerbi and G. Mingione, Gradient estimates for a class of parabolic systems, Duke Mathematical Journal 136 (2007), 285–320.

    Article  MathSciNet  MATH  Google Scholar 

  2. E. Acerbi, G. Mingione and G. A. Seregin, Regularity results for parabolic systems related to a class of non-Newtonian fluids, Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 21 (2004), 25–60.

    MathSciNet  MATH  Google Scholar 

  3. M. D. Amaral and E. V. Teixeira, Free transmission problems, Communications in Mathematical Physics 337 (2015), 1465–1489.

    Article  MathSciNet  MATH  Google Scholar 

  4. D. J. Araújo, G. C. Ricarte and E. V. Teixeira, Geometric gradient estimates for solutions to degenerate elliptic equations, Calculus of Variations and Partial Differential Equations 53 (2015), 605–625.

    Article  MathSciNet  MATH  Google Scholar 

  5. D. J. Araújo, E. V. Teixeira and J. M. Urbano, A proof of the C p′-regularity conjecture in the plane, Advances in Mathematics 316 (2017), 541–553.

    Article  MathSciNet  MATH  Google Scholar 

  6. D. J. Araújo, E. V. Teixeira and J. M. Urbano, Towards the C p′-regularity conjecture in higher dimensions, International Mathematics Research Notices (2018), 6481–6495.

    Google Scholar 

  7. D. J. Araújo and L. Zhang, Optimal c 1,α estimates for a class of elliptic quasilinear equations, Communications in Contemporary Mathematics, to appear, doi:10.1142/S0219199719500147.

  8. A. Attouchi, M. Parviainen and E. Ruosteenoja, C 1,α regularity for the normalized p-Poisson problem, Journal de Mathématiques Pures et Appliquées 108 (2017), 553–591.

    Article  MathSciNet  MATH  Google Scholar 

  9. H.-O. Bae and H. J. Choe, Regularity for certain nonlinear parabolic systems, Communications in Partial Differential Equations 29 (2004), 611–645.

    Article  MathSciNet  MATH  Google Scholar 

  10. V. Bögelein, F. Duzaar and G. Mingione, The regularity of general parabolic systems with degenerate diffusion, Memoirs of the American Mathematical Society 221 (2013).

  11. L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Annals of Mathematics 130 (1989), 189–213.

    Article  MathSciNet  MATH  Google Scholar 

  12. S. Challal, A. Lyaghfouri, J. F. Rodrigues and R. Teymurazyan, On the regularity of the free boundary for quasilinear obstacle problems, Interfaces and Free Boundaries 16 (2014), 359–394.

    Article  MathSciNet  MATH  Google Scholar 

  13. H. J. Choe, Hölder regularity for the gradient of solutions of certain singular parabolic systems, Communications in Partial Differential Equations 16 (1991), 1709–1732.

    Article  MathSciNet  MATH  Google Scholar 

  14. J. V. da Silva and D. dos Prazeres, Schauder type estimates for “flat” viscosity solutions to non-convex fully nonlinear parabolic equations and applications, Potential Analysis 50 (2019), 149–170.

    Article  MathSciNet  MATH  Google Scholar 

  15. J. V. da Silva, R. A. Leitão and G. C. Ricarte, Geometric regularity estimates for fully nonlinear elliptic equations with free boundaries, submitted, https://www.researchgate.net/publication/316283716_Geometric_regularity_ estimates_for_fully_nonlinear_elliptic_equations_with_free_boundaries.

  16. J. V. da Silva, P. Ochoa and A. Silva, Regularity for degenerate evolution equations with strong absorption, Journal of Differential Equations 264 (2018), 7270–7293.

    Article  MathSciNet  MATH  Google Scholar 

  17. J. V. da Silva, J. D. Rossi and A. M. Salort, Regularity properties for p-dead core problems and their asymptotic limit as p → 8, Journal of the London Mathematical Society 99 (2019), 69–96.

    Article  MathSciNet  MATH  Google Scholar 

  18. J. V. da Silva and A. M. Salort, Sharp regularity estimates for quasi-linear elliptic dead core problems and applications, Calculus of Variations and Partial Differential Equations 57 (2018), Art. 83, 24.

  19. J. V. da Silva and E. V. Teixeira, Sharp regularity estimates for second order fully nonlinear parabolic equations, Mathematische Annalen 369 (2017), 1623–1648.

    Article  MathSciNet  MATH  Google Scholar 

  20. E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993.

    Google Scholar 

  21. E. DiBenedetto and A. Friedman, Regularity of solutions of nonlinear degenerate parabolic systems, Journal für die Reine und Angewandte Mathematik 349 (1984), 83–128.

    MathSciNet  MATH  Google Scholar 

  22. E. DiBenedetto and A. Friedman, Addendum to: “Hölder estimates for nonlinear degenerate parabolic systems”, Journal für die Reine und Angewandte Mathematik 363 (1985), 217–220.

    MathSciNet  MATH  Google Scholar 

  23. E. DiBenedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, Journal für die Reine und Angewandte Mathematik 357 (1985), 1–22.

    MathSciNet  MATH  Google Scholar 

  24. E. DiBenedetto, U. Gianazza and V. Vespri, Harnack’s Inequality for Degenerate and Singular Parabolic Equations, Springer Monographs in Mathematics, Springer, New York, 2012.

    Google Scholar 

  25. E. DiBenedetto, J. M. Urbano and V. Vespri, Current issues on singular and degenerate evolution equations, in Evolutionary Equations. Vol. I, Handbook of Differential Equations, North-Holland, Amsterdam, 2004, pp. 169–286.

    Google Scholar 

  26. T. Iwaniec and J. J. Manfredi, Regularity of p-harmonic functions on the plane, Revista Matemática Iberoamericana 5 (1989), 1–19.

    Article  MathSciNet  MATH  Google Scholar 

  27. J. Kinnunen and J. L. Lewis, Higher integrability for parabolic systems of p-Laplacian type, Duke Mathematical Journal 102 (2000), 253–271.

    Article  MathSciNet  MATH  Google Scholar 

  28. N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Graduate Studies in Mathematics, Vol. 96, American Mathematical Society, Providence, RI, 2008.

  29. T. Kuusi and G. Mingione, Gradient regularity for nonlinear parabolic equations, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V 12 (2013), 755–822.

    MathSciNet  MATH  Google Scholar 

  30. T. Kuusi and G. Mingione, The Wolff gradient bound for degenerate parabolic equations, Journal of the European Mathematical Society 16 (2014), 835–892.

    Article  MathSciNet  MATH  Google Scholar 

  31. O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, RI, 1968.

  32. G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing, River Edge, NJ, 1996.

    Book  MATH  Google Scholar 

  33. E. Lindgren and P. Lindqvist, Regularity of the p-Poisson equation in the plane, Journal d’Analyse Mathématique 132 (2017), 217–228.

    Article  MathSciNet  MATH  Google Scholar 

  34. P. Lindqvist, On the time derivative in a quasilinear equation, Skrifter. Det Kongelige Norske Videnskabers Selskab 2 (2008), 1–7.

    MathSciNet  MATH  Google Scholar 

  35. J. Simon, Compact sets in the space L p(0, T; B), Annali di Matematica Pura ed Applicata 146 (1987), 65–96.

    Article  MathSciNet  MATH  Google Scholar 

  36. E. V. Teixeira, Sharp regularity for general Poisson equations with borderline sources, Journal de Mathématiques Pures et Appliquées 99 (2013), 150–164.

    Article  MathSciNet  MATH  Google Scholar 

  37. E. V. Teixeira, Regularity for quasilinear equations on degenerate singular sets, Mathematiche Annalen 358 (2014), 241–256.

    Article  MathSciNet  MATH  Google Scholar 

  38. E. V. Teixeira, Hessian continuity at degenerate points in nonvariational elliptic problems, International Mathematics Research Notices (2015), 6893–6906.

    Google Scholar 

  39. E. V. Teixeira and J. M. Urbano, A geometric tangential approach to sharp regularity for degenerate evolution equations, Analysis & PDE 7 (2014), 733–744.

    Article  MathSciNet  MATH  Google Scholar 

  40. E. V. Teixeira and J. M. Urbano, An intrinsic Liouville theorem for degenerate parabolic equations, Archiv Mathematik 102 (2014), 483–487.

    Article  MathSciNet  MATH  Google Scholar 

  41. J. M. Urbano, The method of intrinsic scaling, Lecture Notes in Mathematics, Vol. 1930, Springer-Verlag, Berlin, 2008.

  42. M. Wiegner, On C α -regularity of the gradient of solutions of degenerate parabolic systems, Annali di Matematica Pura ed Applicata 145 (1986), 385–405.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank Eduardo V. Teixeira and José Miguel Urbano for pointing out several improvements and for their insightful comments and suggestions that much benefited the final outcome of this manuscript. We would also like to thank the anonymous referees for the careful reading and valuable suggestions throughout the paper.

The first author thanks the Department of Mathematics of Universidad de Buenos Aires for providing an excellent working environment during his visit.

J. V. da Silva thanks DM/FCEyN (Universidad de Buenos Aires) for providing a productive working atmosphere.

This work was partially supported by Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET-Argentina) grant PIP GI No. 11220150100036CO, Pronex/CNPq/Funcap (Brazil) 00068.01.00/15 and by FCT-Portugal via the grant SFRH/BPD/92717/2013. J. V. da Silva is a member of CONICET.

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Authors and Affiliations

  1. Department of Mathematics, Universidade da Integração Internacional da Lusofonia Afro-Brasileira - UNILAB, 62785-000, Acarape, Ceará, Brazil

    Marcelo D. Amaral

  2. FCEyN, Department of Mathematics, University of Buenos Aires, Ciudad Universitaria-Pabellón I, Buenos Aires, C1428EGA, Argentina

    João Vitor da Silva

  3. Department of Mathematics, Federal University of Ceará, 60455-760, Fortaleza, Ceará, Brazil

    Gleydson C. Ricarte

  4. CMUC, Department of Mathematics, University of Coimbra, 3001-501, Coimbra, Portugal

    Rafayel Teymurazyan

Authors
  1. Marcelo D. Amaral
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  2. João Vitor da Silva
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  3. Gleydson C. Ricarte
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  4. Rafayel Teymurazyan
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Correspondence to João Vitor da Silva.

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Amaral, M.D., da Silva, J.V., Ricarte, G.C. et al. Sharp regularity estimates for quasilinear evolution equations. Isr. J. Math. 231, 25–45 (2019). https://doi.org/10.1007/s11856-019-1842-1

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