Quasilinear parabolic problems with nonlinear boundary conditions

Definition 2.2.3 Let X and Y be Banach spaces. A family of operators T ⊂ B(X, Y )
is called R-bounded, if there is a constant C > 0 and p ∈ [1, ∞) such that for each
N ∈ N, Tj ∈ T , xj ∈ X and for all independent, symmetric, {−1, 1}-valued random
variables εj on a probability space (Ω, M, µ) the inequality
N�
N�
| εjTjxj| Lp(Ω;Y ) ≤ C| εjxj| Lp(Ω;X)
j=1
is valid. The smallest such C is called R-bound of T , we denote it by R(T ).
j=1
(2.6)
The notion of R-sectorial operators is obtained by replacing bounded with R-bounded in
the definition of sectorial operators.
Definition 2.2.4 Let X be a complex Banach space, and assume A is a sectorial operator
in X. Then A is called R-sectorial if
RA(0) := R{t(t + A) −1 : t > 0} < ∞.
The R-angle φR A of A is defined by means of
where
φ R A := inf{θ ∈ (0, π) : RA(π − θ) < ∞},
RA(θ) := R{λ(λ + A) −1 : |arg λ| ≤ θ}.
The class of R-sectorial operators in X will be denoted by RS(X).
The R-angle of an R-sectorial operator A is well-defined and it is not smaller than the
spectral angle of A, cp. Denk, Hieber and Prüss [29, Definition 4.1].
The following fundamental result, which has been proven in Clément and Prüss [24],
says that the class of R-sectorial operators contains the class of operators with bounded
imaginary powers, provided that the underlying Banach space X belongs to the class
HT , see Section 2.3 for the definition of the latter.
Theorem 2.2.3 Let X be a Banach space of class HT and suppose that A ∈ BIP(X)
with power angle θA. Then A is R-sectorial and φR A ≤ θA.
For sectorial operators A one knows that the powers A α with α ∈ R and |α| < π/φA are
sectorial as well and φA α ≤ |α|φA, see e.g. [29, Thm. 2.3]. It turns out that there is a
corresponding result for the class RS(X).
Proposition 2.2.1 Let X be a complex Banach space. Suppose A ∈ RS(X) and α ∈ R
is such that |α| < π/φ R A . Then Aα is also R-sectorial and φ R A α ≤ |α|φR A .
Proof. In view of A−α = (A−1 ) α , it suffices to consider positive α. In fact, for A ∈ RS(X)
and φ > φR A , the relation
λ(λ + A −1 ) −1 = λA(1 + λA) −1 = A(λ −1 + A) −1 , λ ∈ Σπ−φ,
shows that A−1 ∈ RS(X) and φR A−1 = φR A . So let α ∈ (0, π/φR A ) be fixed. Since
RS(X) ⊂ S(X), it follows that Aα ∈ S(X) with spectral angle φAα ≤ αφA.
Let now φα < π − αφR A and µ ∈ Σφα . Then the function gµ(λ) = µ/(µ + λα ) belongs
to H∞ (Σφ) as long as φα + αφ < π. By means of the extended functional calculus (cf.
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