Quasilinear parabolic problems with nonlinear boundary conditions

Given a function v ∈ H 2 p(R n+1
+ ) we consider the pull-back Θv defined on Ω0 by
Θv(x) = v( ˜ ϑ(x)). Using now the notation x = (x1, . . . , xn+1), the function u = Θv
satisfies
n+1 �
∂xiu(x) = ∂¯xk v( ˜ ϑ(x))∂xi ˜ ϑk(x),
k=1
n+1 �
∂xi∂xj u(x) = ∂¯xk v( ˜ ϑ(x))∂xi∂xj ˜ ϑk(x) +
k=1
n+1 �
k, l=1
∂¯xk ∂¯xl v( ˜ ϑ(x))∂xi ˜ ϑk(x)∂xj ˜ ϑl(x)
for x ∈ Ω0 and i, j = 1, . . . , n + 1. For a differential expression of the form
we thus obtain
(Eu)(x) = −c(x) : ∇ 2 xu(x) + c1(x) · ∇xu(x) + c0(x)u(x), x ∈ Ω0, (4.43)
(Eu)(x) = − (D ˜ ϑ(x)c(x)D ˜ ϑ T (x)) : ∇ 2 ¯xv( ˜ ϑ(x))
+ (D ˜ ϑ(x)c1(x) − D 2 ˜ ϑ(x) : c(x)) · ∇¯xv( ˜ ϑ(x)) + c0(x)v( ˜ ϑ(x)), x ∈ Ω0,
with (D 2 ˜ ϑ(x) : c(x))k = � n+1
i, j=1 cij(x)∂xi ∂xj ˜ ϑk(x), k = 1, . . . , n + 1. So applying the
push-forward operator Θ −1 to the function Eu on Ω0 gives
Θ −1 Eu = (Θ −1 EΘ)Θ −1 u = E ˜ ϑ v,
where E ˜ ϑ := Θ −1 EΘ is the second order differential operator
(E ˜ ϑ ˜
w)(¯x) = −c
ϑ(¯x) 2
: ∇¯xw(¯x) + c ˜ ϑ
1(¯x) · ∇¯xw(¯x) + c ˜ ϑ
0(¯x)w(¯x), ¯x ∈ R n+1
+ , (4.44)
with coefficients
c ˜ ϑ (¯x) = (D ˜ ϑ c D ˜ ϑ T ) ◦ ˜ ϑ −1 (¯x), c ˜ ϑ1(¯x) = (D ˜ ϑc1 − D 2 ˜ ϑ : c) ◦ ˜ ϑ −1 (¯x),
c ˜ ϑ
0(¯x) = c0( ˜ ϑ −1 (¯x)), ¯x ∈ R n+1
+ , (4.45)
see also [69, Section 5].
Observe that the preceeding formulas are also valid for functions on J × Ω0 resp.
J × R n+1
+ and differential operators (4.43) resp. (4.44) with time-dependent coefficients.
In view of (4.45), it follows in particular that for an operator A(t, x, Dx) of the form
(4.41) and satisfying the smoothness and ellipticity conditions in (H2) and (H3) with
Ω replaced by Ω0, the transformed operator A ˜ ϑ (t, ¯x, D¯x) defined on J × R n+1
+ enjoys
the same properties, the ellipticity constant c0 appearing in (H3) remaining unchanged.
Further, since D ˜ ϑ ≡ 1 and the derivatives of ˜ ϑ and ˜ ϑ−1 up to order 2 are bounded, the
change of variable formula for the Lebesgue integral shows that Θ induces isomorphisms
Θ (p) : Hm p (R n+1
+ ) → Hm p (Ω0) for each p ∈ (1, ∞) and m = 0, 1, 2.
Proof of Theorem 4.3.1. We begin this time with the sufficiency part. The overall
plan can roughly described as follows. With the aid of localization w.r.t. space and
the coordinate transformations discussed above, problem (4.40) is reduced to a finite
number of related problems on Rn+1 and R n+1
+ , respectively. For these problems, solution
operators are available thanks to the Theorems 4.1.1, 4.2.3, and 4.2.4. So the local
equations can be solved and, by summing over all ’local solutions’, we obtain a fixed
point equation for v of the form v = v0 + R(v) =: G(v), where v0 is determined by
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