- EXISTENCE OF A MINIMIZER FOR THE ENTROPY 25
STEP 2. Woo is a weak solution of (17. 77). Since w 00 is a minimizer
in W^1 ,^2 of 1-l (g, w) subject to the constraint JM w^2 dμ = 1, for any W^1 ,^2
function</>: M-+ lPi. such that JM w 00 ¢dμ = 0, we have(17.76)0 =! 's=O 1-l (g, W^00 + s
=! ls=O JM ( 4 J\7 (woo+ s¢)J
2
) dμ+ :1 8 { (R-log(woo+s¢)^2 -ilog(47r)-n)(w 00 +s¢)^2 dμ
s=O JM= 2 JM (4\7w 00 , \7¢) + ( R-log (w~) - i log (47r) - n )w 00 ¢-w 00 ¢ )dμ.
That is, by definition w 00 is a weak solution to the following second-order
elliptic equation(17.77) -4~w 00 +Rw 00 -w 00 log (w~)-(i log(47r) + n) W 00 = μ(g, l)w 00 •
The constant μ(g, 1) is determined by (17. 75) and by substituting ¢ = w 00
in (17.76).
STEP 3. w 00 is a positive C^00 minimizer. Define2n
Since w 00 E Ln-2, by (17.116) below we have P (w 00 ) E V for any p E
[1, n^2 :: 2 ) (here ;;:: 2 ~ 00 if n = 2), SO that W 00 E W^2 ,P by the standard
(interior) V estimate for weak solutions to second-order elliptic equations
(see Theorem 9.11 in [71]). Bootstrapping, we obtain Woo E W^2 ,q for all
q E [1, oo). By the Sobolev embedding theorem, this implies that w 00 E C^1 '°'
for some a E (0, 1).Since w 00 2:: 0 everywhere and w 00 > 0 somewhere (since JM w~dμ = 1),
by the strong maximum principle for weak solutions of (17. 77) (see Lemma
17 .26 below), we have w 00 > 0 everywhere on M. Therefore w 00 log w 00 E
C^1 '°' so that'
4~W 00 = P ( w 00 ) E C^1 '°'.By the regularity theory for weak solutions of the Poisson equation, w 00 is a
classical solution and we may apply Schauder theory to conclude that w 00 E
Ck,a for all k E N. Hence w 00 is C^00 , so that fr ~ -i log ( 47r) - 2 log w 00
is a C^00 minimizer of K (g, · , 1). This completes the proof of Proposition
17.24. 0
To conclude this subsection, we prove the following result, which was
used in the proof of Proposition 17.24 (seep. 112 of [161]).