Conformally Invariant Variational Problems. - SAM

Bringing altogether (VIII.15), (VIII.18), (VIII.19), (VIII.21) et
(VIII.22) produces
∫ ∫
|A∇u| 2 ≤ 3δ 2 |A∇u| 2
B δr (p)
B r (p)
+C 1 ε 0
∫
B r (p)
|∇u| 2
(VIII.23)
Using the hypotheses that A and A −1 are bounded in L ∞ , it
follows from (VIII.23) that for all 1 > δ > 0, there holds the
estimate
∫
∫
|∇u| 2 ≤ 3‖A −1 ‖ ∞ ‖A‖ ∞ δ 2 |∇u| 2
B δr (p)
+C 1 ‖A −1 ‖ ∞ ε 0
∫
B r (p)
B r (p)
|∇u| 2 .
(VIII.24)
Next, we choose ε 0 and δ strictly positive, independent of r et
p, and such that
3‖A −1 ‖ ∞ ‖A‖ ∞ δ 2 +C 1 ‖A −1 ‖ ∞ ε 0 = 1 2
.
B δr (p)
B r (p)
Iterating this inequality as in the previous regularity proofs
yields the existence of some constant α > 0 for which
∫
sup ρ −2α |∇u| 2 < +∞ .
p∈B 1/2 (0) , 0