Page 136 - Textos de Matemática Vol. 34

126 JOSE´ MIGUEL URBANO
Theorem 2.2. Let u ∈ H1(Br) be a weak solution of (2.1) in Br. There exists a
constant γ = γ(N,λ,Λ) ∈ 1,1 such that 2
By induction,
osc(u,r/2)≤γosc(u,r) . osc(u,2−kr)≤γk osc(u,r); k=1,2,...
Now, for ρ < r, we can take k such that 2−k−1 r < ρ ≤ 2−k r to obtain ρα
osc(u,ρ)≤C r osc(u,r)
with α = − ln γ ∈ (0, 1), which gives a H¨older modulus of continuity and
ln2
Theorem 2.3 (DeGiorgi-Nash). Every weak solution of (2.1) is locally H¨older continuous.
3. Regularity through intrinsic scaling: a few examples
As we have seen, for strongly elliptic equations the technique used to obtain results on the continuity of weak solutions (including quantitative information on their modulus of continuity) is based on energy estimates and this is also true for parabolic equations. These estimates are essential to set forward the iterative argument consisting of showing that, for every point in the domain, we can find a sequence of nested and shrinking cylinders such that, as the cylinders shrink to the point, the essential oscillation of the solution in the cylinders converges to zero. It is crucial to the reasoning that the equation is strongly elliptic/parabolic so that the integral norms appearing in the estimates are homogeneous.
In the case of degenerate or singular equations this homogeneity is lost and different powers appear in the energy estimates. Intrinsic scaling consists in looking at the equation in its own geometry, i.e., in a geometry dictated by its singular or degenerate structure. This amounts to rescale the standard parabolic cylinders by a factor depending on the oscillation of the solution. The procedure allows for the recovering of the homogeneity in the energy estimates if written over these rescaled cylinders. Let us analyze a few examples.
Given a point x0 ∈ RN, denote by Kρ(x0) the N-dimensional cube with centre at x0 and wedge 2ρ:

Kρ(x0):= x∈RN : max |xi −x0i|<ρ ; 1≤i≤N
given a point (x0,t0) ∈ RN+1, the cylinder of radius ρ and height τ > 0 is (x0, t0) + Q(τ, ρ) := Kρ(x0) × (t0 − τ, t0) .
To simplify, we assume throughout (by translation) that (x0 , t0 ) = (0, 0).