Preprint series: 06-06 , Reports on Analysis

MSC:

26A33 Fractional derivatives and integrals

45A05 Linear integral equations

45D05 Volterra integral equations, See also {34A12}

Abstract: We prove a priori estimates for nonnegative supersolutions of
fractional differential equations of the form $\partial_t^\alpha
(u-u_0)+\mu u=f$, $u(0)=u_0$, with $\alpha\in(0,1)$. As a main
result, we establish for such functions a weak Harnack inequality
with critical exponent $1/(1-\alpha)$, which is shown to be
optimal. In addition, we obtain an $L_p$-estimate of Moser type
and show that positive supersolutions satisfy certain
$\log$-estimates; the latter plays a crucial role in connection
with an abstract lemma of Bombieri and Giusti, which is an
extremely useful tool to prove Harnack type estimates for a wide
class of elliptic and parabolic problems. Therefore,
the results obtained are also of preliminary character with
regard to a corresponding theory for fractional evolution
equations of the form $\partial_t^\alpha (u-u_0)-L u=f$, where
$L$ stands for a uniformly elliptic operator of second order.

Keywords: a priori estimates, local estimates, weak Harnack inequality, fractional derivative, non-local operator, Volterra equation

Upload: 2006-05-16