- EXISTENCE AND CONVERGENCE 91
books on PDE, such as Moser [276], Morrey [274], Gilbarg and Trudinger
[155], Han and Lin [195] (e.g., see Theorem 4.15 on p. 83 of [195]).THEOREM 2.61 (Harnack inequality). Let u : B(3Ro) -+ JR be a C^2
function such that
1
Ao ( <5°'13) :'S: ( u°'13) :'S: Ao ( <5°'13)for some Ao E [1, oo ). Suppose that a nonnegative function v E W^2 ,^2 (B(3Ro))
and g E Lq(B(3Ro)), for some q > m/2, satisfy/J.uv :'S: g
in the weak sense in B(3Ro), where !J.u denotes the Laplacian with respectto the metric ua/3· Then for any 0 < () :'S: r < 1 and 0 < p < m~ 2 , there
exists a constant C = C(p, q, 2n, Ao,(), r) < oo such that for any p::; 2Ro,
1.(2.84) (~ { V (y)P dy) P :'S: C ( inf V + p
2
-rr: ll!JllLq(B(p))) ·
pm J B(Tp) B(()p)REMARK 2.62. The reason for why we can apply this theorem to( ucx13) = (gcx/3) = (g~/3 + 'Pa/3)
is that the C^2 -estimate for <p yields (2.78).Note that !J.u (M(2) - w) :'S: -h'Y"Y and -h'Y"Y E Lq for all q :'S: oo (e.g.,q < oo ), so we may apply the above theorem to M(2)-w (for example, with
m = 2n, p = 2R,^6 and()= r = ~' so that ()p = rp = R) to obtain for any
q > n and 0 < p < n::_l that there exists C = C(p, q, n, A) < oo such that
1(R
; { (M(2) - w (y))Pdy) P
n jB(R)(2.85) :'S: C (M(2) - M(l) + R
2
(q;;-n) llh'Y"fl1Lq(B(2R)))since infB(R) (-w) = M(l).
On the other hand, the concavity of F implies
{)F
F(ui;(x)) :'S: F(ui3(y)) + {)p.~ (uiJ(Y))(ui;(x) - uiJ(Y)).
iJ
Namely we have(2.86)
The following linear algebra fact enables us to estimate w(R).(^6) Note that R:::; Ro.