Page 135 - Textos de Matemática Vol. 34

THE GEOMETRY OF SOME NONLINEAR PDES 125
and can prove, by induction, that, for some γ > 1,
(2.6) ψ(kn,rn)≤ ψ(k0,r0) , ∀n=0,1,2,...
γn
if k is chosen sufficiently large. In fact, it is trivial that it holds for n = 0; now
suppose it holds for n − 1 and write 2
ψ(k ,r )1+ 2 ≤ 0 0
γn−1
= ψ(k0,r0)N ψ(k0,r0) .
ψ(kn−1, rn−1)1+ N
N
2
2n −(1+ 2 ) γNNγ
n
From (2.5) we obtain
ψ(k ,r ) ≤ 2Cγ1+ 2
2 n(1+ 2 ) ψ(k0,r0)N 2 N ψ(k0,r0)
nn N 2 2n n kNγNγ
andchoosefirstγ>1suchthatγ2 =21+2 andthenklargeenoughsothat 2
NN
2Cγ1+2 ψ(k0,r0)N ≤1 ⇐ k=C∗ψ(k,r), N200
kN
Finally, let n → ∞ in (2.6) to get ψ(k, 1) ≤ 0, i.e.,
where C∗ ≡ C∗(N, λ, Λ).
Hence
2 ∥(u−k)+∥L2(B1) =0.
2
sup u+ ≤ C∗ ∥u+∥L2(B1) . B1
2
Using a dilation argument, this estimate can be refined; indeed, for any θ ∈ (0, 1) and p > 1, it holds
sup u+ ≤ C(N,λ,Λ) ∥u+∥Lp(B1) . Bθ (1 − θ)n/p
The same type of reasoning gives similar conclusions concerning u− and the result follows.
The basic general idea to obtain results concerning the continuity of a solution of a PDE at a point consists in estimating its oscillation in a nested sequence of concentric balls (cylinders in the parabolic case), centered at the point, and showing that it converges to zero as the balls shrink to the point. If this can be measured quantitatively it gives a modulus of continuity. Denote the oscillation of a function u in Br with osc (u, r). A further analysis, which uses the previous theorem, leads to