Quasilinear parabolic problems with nonlinear boundary conditions

µ ∈ Σφα }. Let N ∈ N, Tj ∈ T , xj ∈ X and suppose that εj are independent, symmetric,
{−1, 1}-valued random variables on a probability space (Ω, M, µ), j = 1, . . . , n. Let
further Tε, j ∈ Tε be the approximation of Tj, that is, for Tj = gµj (A) we put Tε, j =
gµj (Aε). Also, we choose for each xj a sequence {xj, k} ∞ k=1 ⊂ D(A) ∩ R(A) such that
xj, k → xj as k → ∞. By uniform R-boundedness of Tε, we may then estimate
N�
N�
N�
| εjTjxj| Lp(Ω;X) ≤ | εjTε, j xj| Lp(Ω;X) + | εj(Tj − Tε, j)xj, k| Lp(Ω;X)
j=1
j=1
+ |
≤ C |
j=1
N�
εj(Tj − Tε, j)(xj − xj, k)| Lp(Ω;X)
j=1
N�
εjxj| Lp(Ω;X) +
j=1
N�
|(Tj − Tε, j)xj, k|X + 2C
j=1
N�
|xj − xj, k|X,
j=1
(2.8)
where C does not depend on ε, j, k. Now let ε → 0+ in (2.8) with k being fixed. This
makes the second summand disappear. The third one vanishes if we then send k → ∞.
It remains the desired inequality expressing R-boundedness of {gµ(A) : µ ∈ Σφα } for
each φα < π − αφR A . �
Connecting the concept of R-boundedness to the H ∞ -calculus, leads to the notion of
operators with R-bounded functional calculus.
Definition 2.2.5 Let X be a complex Banach space and suppose that A ∈ H ∞ (X). The
operator A is said to admit an R-bounded H ∞ -calculus if the set
{f(A) : f ∈ H ∞ (Σθ), |f| θ ∞ ≤ 1}
is R-bounded for some θ > 0. We denote the class of such operators by RH∞ (X) and
define the RH∞-angle φR∞ A of A as the infimum of such angles θ.
One important application of such operators concerns the joint functional calculus of
sectorial operators, see Section 2.4.
2.3 Sums of closed linear operators
Let X be a Banach space, A, B closed linear operators in X, and consider the problem
Ax + Bx = y. (2.9)
Given y ∈ X one seeks a unique strict solution x of (2.9) in the sense that x ∈ D(A) ∩
D(B), that is x possesses the regularity induced by A as well as that coming from B.
In this situation we say that the solution has maximal regularity. Furthermore it is
desirable to have an a priori estimate of the form
where C does not depend on x.
|Ax| + |Bx| ≤ C|Ax + Bx| for all x ∈ D(A) ∩ D(B), (2.10)
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