Quasilinear parabolic problems with nonlinear boundary conditions

Theorem 4.3.1 Let 1 < p < ∞, J = [0, T ], and Ω ⊂ R n+1 be a domain with compact
C 2 -boundary Γ which decomposes according to Γ = ΓD ∪ ΓN and dist (ΓD, ΓN) > 0.
Suppose the assumptions (H1)-(H4) are satisfied. Then (4.40) admits a unique solution
in the space
Z := H α p (J; Lp(Ω)) ∩ Lp(J; H 2 p(Ω))
if and only if the functions f, g, h are subject to the following conditions.
(i) f ∈ H α p (J; Lp(Ω));
(ii) g ∈ B
(iii) h ∈ B
1
α(1− 2p )
pp
1 1
α( − 2 2p )
pp
(iv) f|t=0 ∈ B
2
2− pα
pp
2− 1
p
(J; Lp(ΓD)) ∩ Lp(J; Bpp (ΓD));
1− 1
p
(J; Lp(ΓN)) ∩ Lp(J; Bpp (ΓN));
(Ω), if α > 1
p ;
1 1
2(1− − α pα )
pp
(v) ∂tf|t=0 ∈ B (Ω), if α > 1 + 1
p ;
(vi) f|t=0 = g|t=0 on ΓD, if α > 2
2p−1 ;
(vii) B(0, x, Dx)f|t=0 = h|t=0 on ΓN, if α > 2
p−1 ;
(viii) ∂tf|t=0 = ∂tg|t=0 on ΓD, if α > 1 + 3
2p−1 .
Before proving Theorem 4.3.1 we recall some general properties of variable transformations.
Let Ω ⊂ R n+1 be a domain with compact C 2 -boundary Γ and x0 ∈ Γ. Without
restriction of generality, we may assume that x0 = 0 and that n(x0) = (0, . . . , 0, −1);
this can always be achieved by a composition of a translation and a rotation in R n+1 . It
is easy to see that such affine mappings of R n+1 onto itself leave invariant all function
spaces under consideration (i.e. Lp, Z, and the regularity classes of the data and of
the coefficients), and they also preserve ellipticity (including the ellipticity constant in
(H3)) as well as normality. Continuing, by definition of a C 2 -boundary, there exist an
open neighbourhood U = U1 × U2 ⊂ R n+1 of x0 with U1 ⊂ R n and U2 ⊂ R as well as a
function ζ ∈ C 2 (U1) such that
Define ϑ : Ū → Rn+1 by
Γ ∩ U = {x = (x ′ , y) ∈ U : y = ζ(x ′ )},
Ω ∩ U = {x = (x ′ , y) ∈ U : y > ζ(x ′ )}.
ϑk(x) = x ′ k if k = 1, . . . , n and ϑn+1(x) = y − ζ(x ′ ). (4.42)
Clearly, ϑ ∈ C 2 (U, R n+1 ) is one-to-one and satisfies Ω ∩ U = {x ∈ U : ϑn+1(x) > 0}
as well as Γ ∩ U = {x ∈ U : ϑn+1(x) = 0}. By extending ζ to a function ˜ ζ ∈ C 2 (R n )
with compact support and defining ˜ ϑ by (4.42) with ζ being replaced by ˜ ζ, we get a
C 2 -diffeomorphism ˜ ϑ of R n+1 onto itself extending ϑ and satisfying ˜ ϑ(x) = x for large
values of |x|. Also, ˜ ϑ is a C 2 -diffeomorphic mapping from Ω0 := {x ∈ R n+1 : y > ˜ ζ(x ′ )}
onto R n+1
+ . For the Jacobian D ˜ ϑ(x), one obtains
D ˜ �
In 0
ϑ(x) =
−∇x ′ ζ(x ˜ ′ ) 1
�
, x ∈ R n+1 ,
which entails det D ˜ ϑ(x) = 1 for all x ∈ R n+1 . Notice also that D ˜ ϑ(0) = In+1.
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