Page 137 - Textos de Matemática Vol. 34

THE GEOMETRY OF SOME NONLINEAR PDES 127
3.1. The degenerate p-Laplace equation. Consider the quasilinear parabolic equation, with principal part in divergence form,
(3.1) ut−div|∇u|p−2∇u=0 in ΩT ; p>2.
The equation is degenerate in the space part due to the vanishing of its modulus of ellipticity |∇u|p−2 at points where |∇u| = 0.
ConsideracylinderQ(τ,ρ)⊂ΩT andlet0≤ζ≤1beapiecewisesmoothcutoff function in Q(τ, ρ) such that
|∇ζ|<∞ and ζ(x,t)=0, x̸∈Kρ ≡Kρ(0).
Proposition 3.1. Let u be a local weak solution of (3.1) and k ∈ R. There exists a constant C ≡ C(p) > 0 such that for every cylinder Q(τ, ρ) ⊂ ΩT ,
2p0p
sup (u−k)± ζ dx+ −τ <t<0 Kρ ×{t}
|∇(u−k)±ζ| dxdt −τ Kρ
2p0pp
≤ (u−k)± ζ dx + C (u−k)± |∇ζ| dxdt
Kρ×{−τ} −τ Kρ 0
+p (u−k)2±ζp−1∂tζdxdt. −τ Kρ
These energy estimates are not homogeneous in the space and time parts due to the presence of the power p > 2 and, to go about this difficulty, we consider a geometry dictated by the structure of the equation. Let R > 0 be such that Q(R2, R) ⊂ ΩT , define
μ+ :=esssup u ; Q(R2 ,R)
and construct the cylinder
μ− :=essinf u ; Q(R2 ,R)
ω:=essosc u=μ+−μ− Q(R2 ,R)
p
Q(a0R ,R)≡KR(0)×(−a0R ,0)
ω 2−p a0 = 2λ ,
p
with
where λ > 1 is to be fixed depending only on the data. Note that for p = 2, i.e., in the non-degenerate case, these are the standard parabolic cylinders reflecting the natural homogeneity between the space and time variables.
We assume, without loss of generality, that (3.2) R< ω
2λ
because if this doesn’t hold there is nothing to prove since the oscillation is com-
parable to the radius. Now, (3.2) implies the inclusion Q(a0Rp, R) ⊂ Q(R2, R)