Hölder and Lipschitz continuity of the solutions to parabolic equations of the non-divergence type

Seiichiro Kusuoka

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

We consider time-inhomogeneous, second-order linear parabolic partial differential equations of the non-divergence type, and assume the ellipticity and the continuity on the coefficient of the second-order derivatives and the boundedness on all coefficients. Under the assumptions, we show the Hölder continuity of the solution in the spatial component. Furthermore, additionally assuming the Dini continuity of the coefficient of the second-order derivative, we have the better continuity of the solution. In the proof, we use a probabilistic method, in particular the coupling method. As a corollary, under an additional assumption we obtain the Hölder and Lipschitz continuity of the fundamental solution in the spatial component.

Original languageEnglish
Pages (from-to)1-26
Number of pages26
JournalJournal of Evolution Equations
DOIs
Publication statusAccepted/In press - Nov 9 2016

Fingerprint

Lipschitz Continuity
Parabolic Equation
Second-order Derivatives
Coefficient
Coupling Method
Ellipticity
Probabilistic Methods
Linear partial differential equation
Parabolic Partial Differential Equations
Fundamental Solution
Boundedness
Corollary

Keywords

  • Coupling method
  • Diffusion
  • Fundamental solution
  • Hölder continuous
  • Lipschitz continuous
  • Parabolic partial differential equation
  • Stochastic differential equation

ASJC Scopus subject areas

  • Mathematics (miscellaneous)

Cite this

Hölder and Lipschitz continuity of the solutions to parabolic equations of the non-divergence type. / Kusuoka, Seiichiro.

In: Journal of Evolution Equations, 09.11.2016, p. 1-26.

Research output: Contribution to journalArticle

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