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 language | English |
|---|---|
| Pages (from-to) | 1-26 |
| Number of pages | 26 |
| Journal | Journal of Evolution Equations |
| DOIs | |
| Publication status | Accepted/In press - Nov 9 2016 |
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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 journal › Article
}
TY - JOUR
T1 - Hölder and Lipschitz continuity of the solutions to parabolic equations of the non-divergence type
AU - Kusuoka, Seiichiro
PY - 2016/11/9
Y1 - 2016/11/9
N2 - 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.
AB - 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.
KW - Coupling method
KW - Diffusion
KW - Fundamental solution
KW - Hölder continuous
KW - Lipschitz continuous
KW - Parabolic partial differential equation
KW - Stochastic differential equation
UR - http://www.scopus.com/inward/record.url?scp=84994469265&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84994469265&partnerID=8YFLogxK
U2 - 10.1007/s00028-016-0373-z
DO - 10.1007/s00028-016-0373-z
M3 - Article
AN - SCOPUS:84994469265
SP - 1
EP - 26
JO - Journal of Evolution Equations
JF - Journal of Evolution Equations
SN - 1424-3199
ER -