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


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
$$ \frac{\partial u}{\partial t} - F(x, t, D^{2} u) = f(x, t) \quad \text{in} \quad Q_{1} = B_{1} \times (-1, 0], $$
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.


Fully nonlinear parabolic equations Flat viscosity solutions Schauder type estimates 

Mathematics Subject Classification (2010)



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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.


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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

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