Advertisement

Potential Analysis

, Volume 50, Issue 2, pp 149–170 | Cite as

Schauder Type Estimates for “Flat” Viscosity Solutions to Non-convex Fully Nonlinear Parabolic Equations and Applications

  • João Vítor da SilvaEmail author
  • Disson dos Prazeres
Article

Abstract

In this manuscript we establish Schauder type estimates for viscosity solutions with small enough oscillation to non-convex fully nonlinear second order parabolic equations of the following form
$$ \frac{\partial u}{\partial t} - F(x, t, D^{2} u) = f(x, t) \quad \text{in} \quad Q_{1} = B_{1} \times (-1, 0], $$
(Eq)
provided that the source f and the coefficients of F are Dini continuous functions. Furthermore, for problems with merely continuous data, we prove that such solutions are parabolically C1,Log-Lip smooth. Finally, we put forward a number of applications consequential of our estimates, which include a partial regularity result and a theorem of Schauder type for classical solutions.

Keywords

Fully nonlinear parabolic equations Flat viscosity solutions Schauder type estimates 

Mathematics Subject Classification (2010)

35B65 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The first author was partially supported by Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET-Argentina), Capes and CNPq from Brazil. The second author was partially supported by Capes-Fapitec and CNPq from Brazil, and Proyecto Basal PFB 03 - Chile. We would like to thank the Department of Mathematics at Universidade Federal do Ceará, FCEyN - Universidad de Buenos Aires and Center for Mathematical Modeling - University of Chile for fostering a pleasant and productive scientific atmosphere, which has benefited a lot the final outcome of this current project. The authors would like to thank Eduardo Teixeira for his constant encouragement and support, as well as for several insightful suggestions throughout the elaboration of this manuscript. The authors are also grateful to the anonymous referee for her/his careful review and for pointing out a number of improvements that enormously benefited the final version of the article.

References

  1. 1.
    Armstrong, S., Silvestre, L., Smart, C.: Partial regularity of solutions of fully nonlinear, uniformly elliptic equations. Comm. Pure Appl. Math 65, 1169–1184 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Caffarelli, L.A., Stefanelli, U.: A counterexample to C 2,1 regularity for parabolic fully nonlinear equations. Comm. Partial Differential Equations 33(7–9), 1216–1234 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, Y.Z., Zou, X.: Fully nonlinear parabolic equations and the dini condition. Acta Math. Sin-English Series 18(3), 473–480 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Crandall, M.G., Kocan, M., Swiech, A.: L p-theory for fully nonlinear uniformly parabolic equations. Comm. Partial Differential Equations 25(11–12), 1997–2053 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Daniel, J.-P.: Quadratic expansions and partial regularity for fully nonlinear uniformly parabolic equations. Calc. Var. Partial Differential Equations 54, 183–216 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    da Silva, J.V., Teixeira, E.V.: Sharp regularity estimates for second order fully nonlinear parabolic equations. Math. Ann. 369(3–4), 1623–1648 (2017).  https://doi.org/10.1007/s00208-016-1506-y MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Huisken, G.: The volume preserving mean curvature flow. J. für die Reine und Angewandte Mathematik 382, 35–48 (1987)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Il’in, A.M: On parabolic equations whose coefficients do not satisfy the Dini condition. Mat. Zametki 1(1), 71–80 (1967)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Kovats, J.: Dini-Campanato spaces and applications to nonlinear elliptic equations. Electron. J. Diff. Equa. 1999(37), 1–20 (1999)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Kruzhkov, S.N.: Estimates for the highest derivatives of solutions of elliptic and parabolic equations with continuous coefficients. Mat. Zametki 2(5), 549–560 (1967)MathSciNetGoogle Scholar
  11. 11.
    Krylov, N.: Boundedly inhomogeneous elliptic and parabolic equations in a domain. Izv. Akad. Nak. SSSR. Ser. Mat. 47, 75–108 (1983). English transl. in Math USSR Izv., vol. 22, no. 1, pp. 67–97, 1984MathSciNetGoogle Scholar
  12. 12.
    Krylov, N.: Lectures on Elliptic and Parabolic Equations in Hölder Spaces. Graduate Studies in Mathematics, vol. 12. American Mathematical Society, Providence (1996). xii+ 164 pp.Google Scholar
  13. 13.
    Krylov, N., Safonov, M.: A certain properties of solutions of parabolic equations with measurable coefficients. Math. USSR Izv. 16(1), 151–164 (1981)CrossRefzbMATHGoogle Scholar
  14. 14.
    Lieberman, G.M.: Second Order Parabolic Differential Equations. World Scientific Publishing Co Inc., River Edge (1996)CrossRefzbMATHGoogle Scholar
  15. 15.
    Nadirashvili, N., Vlăduţ, S.: Nonclassical solutions of fully nonlinear elliptic equations. Geom. Funct Anal. 17(4), 1283–1296 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Nadirashvili, N., Vlăduţ, S.: Singular viscosity solutions to fully nonlinear elliptic equations. J. Math Pure Appl. (9) 89(2), 107–113 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Nadirashvili, N., Vlăduţ, S.: Nonclassical solutions of fully nonlinear elliptic equations II. Hessian equations and octonions. Geom. Funct. Anal. 21, 483–498 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Nadirashvili, N., Vlăduţ, S.: Octonions and singular solutions of Hessian elliptic equations. Geom. Funct. Anal. 21(2), 483–498 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Nadirashvili, N., Vlăduţ, S.: Singular solutions of Hessian elliptic equations in five dimensions. J. Math. Pures Appl. 100, 769–784 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Prazeres, D., Teixeira, E.V.: Asymptotics and regularity of flat solutions to fully nonlinear elliptic problems. Ann. Sc. Norm. Super. Pisa Cl. Sci. XV, 485–500 (2016)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Savin, O.: Small perturbation solutions for elliptic equations. Comm. Partial Differential Equations 32(4–6), 557–578 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sheng, W., Wang, X.-J.: Regularity and singularity in mean curvature flow. Trends in Partial Differential Equations. Adv. Lect. Math. 399–436 (2010)Google Scholar
  23. 23.
    Teixeira, E.V.: Universal moduli of continuity for solutions to fully nonlinear elliptic equations. Arch. Rational Mech. Anal. 211(3), 911–927 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Teixeira, E.V.: Geometric regularity estimates for elliptic equations. Proc. MCA Contemp. Math. 656(2016), 185–204 (2013)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Teixeira, E.V., Urbano, J.M.: A geometric tangential approach to sharp regularity for degenerate evolution equations. Anal. PDE 7(3), 733–744 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Tian, G., Wang, X.-J.: A priori estimates for fully nonlinear parabolic equations. Int. Math. Res Notices 2012, 1–21 (2012)CrossRefGoogle Scholar
  27. 27.
    Wang, L.: On the regularity theory of fully nonlinear parabolic equations: I. Comm. Pure Appl. Math. XLV, 27–76 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Wang, L.: On the regularity theory of fully nonlinear parabolic equations: II. Comm. Pure Appl. Math. XLV, 141–178 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Wang, Y.: Small perturbation solutions for parabolic equations. Indiana Univ. Math. J. 62(2), 671–698 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Facultad de Ciencias Exactas y Naturales, Departamento de MatemáticaUniversidad de Buenos AiresBuenos AiresArgentina
  2. 2.Departamento de Matemática, Facultad de Ciencias Exactas y NaturalesUniversidad de Buenos AiresBuenos AiresArgentina

Personalised recommendations