Chapter 25. Estimates of the Heat Equation for Evolving Metrics
You talk about things that nobody cares.- From "Sweet Emotion" by Aernsmith
In this chapter we discuss estimates for heat-type equations with respect
to evolving Riemannian metrics. In particular, we consider mean value-type
inequalities and differential Harnack estimates. Generally, the proofs of the
former are based on integral estimates and the proofs of the latter rely on
applications of the maximum principle to gradient-type quantities.
In §1 we present the mean value inequality for subsolutions to heat-type
equations using Moser iteration.
In §2 we discuss the Li-Yau differential Harnack estimate for positive
solutions to heat-type equations.
with respect to evolving metrics 1. Mean value inequality for solutions of heat-type equations
with respect to evolving metrics
For a second-order elliptic or parabolic equation, the mean value inequal-
ity bounds a subsolution in a smaller ball in terms of its integral in a larger
ball. This fundamental inequality, which we now consider for solutions of
heat-type equations with respect to evolving metrics, is useful for obtaining
upper estimates for the heat kernel (see §2 in Chapter 26).
1.1. Statement of the parabolic mean value inequality for an
evolving .metric.
First, in the fixed metric case, we may formalize the parabolic mean
value property as follows. Recall B (p, r) = {x EM : d (x,'p) < r }.
DEFINITION 25.1. We say that a complete Riemannian manifold (Mn, g)
satisfies the parabolic mean value inequality up to time T E (0, oo]
with constant C < oo if for any (p, t) E M x [O, T) and any nonnegative
subsolution f to the heat equation on P = B (p, Vt) x [O, t], i.e.,
we have
of< ~f
8t - 'f (p, t)::; Vo~P) l f dμdt,
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