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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Schauder estimates for equationswith fractional derivatives


Authors: Ph. Clément, G. Gripenberg and S-O. Londen
Journal: Trans. Amer. Math. Soc. 352 (2000), 2239-2260
MSC (2000): Primary 35K99, 45K05
Published electronically: February 14, 2000
MathSciNet review: 1675170
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Abstract:

The equation \begin{equation*}D^\alpha_t (u-h_1) + D^\beta_x(u-h_2) =f,\quad 0< \alpha,\beta< 1, \quad t,x \geq 0,\tag{$*$ } \end{equation*}where $D^\alpha_t$ and $D^\beta_x$ are fractional derivatives of order $\alpha$ and $\beta$ is studied. It is shown that if $f=f(\underline{t}, \underline{x})$, $h_1=h_1(\underline{x})$, and $h_2=h_2(\underline{t})$ are Hölder-continuous and $f(0,0) =0$, then there is a solution such that $ D^\alpha_t u$ and $D^\beta_x u$ are Hölder-continuous as well. This is proved by first considering an abstract fractional evolution equation and then applying the results obtained to ($*$). Finally the solution of ($*$) with $f=1$ is studied.


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Additional Information

Ph. Clément
Affiliation: Faculty of Technical Mathematics, and Informatics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands
Email: clement@twi.tudelft.nl

G. Gripenberg
Affiliation: Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, FIN-02015 HUT, Finland
Email: gustaf.gripenberg@hut.fi

S-O. Londen
Affiliation: Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, FIN-02015 HUT, Finland
Email: stig-olof.londen@hut.fi

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02507-1
PII: S 0002-9947(00)02507-1
Keywords: Fractional derivative, maximal regularity, Schauder estimate, H\"older continuity, fundamental solution, integro-differential equation
Received by editor(s): March 20, 1997
Received by editor(s) in revised form: September 29, 1997
Published electronically: February 14, 2000
Additional Notes: The third author acknowledges the partial support of the Nederlandse organisatie voor wetenschappelijk onderzoek (NWO)
Article copyright: © Copyright 2000 American Mathematical Society