Quasilinear parabolic problems with nonlinear boundary conditions

Given φ, φ ′ ∈ (0, π] we denote by H ∞ (Σφ × Σφ
′) the Banach algebra of all bounded
φ′
holomorphic scalar-valued functions on Σφ × Σφ ′ equipped with the norm |f|φ, ∞
sup{|f(λ, λ ′ )| : |arg λ| < φ, |arg λ ′ | < φ ′ }, and we put H0(Σφ × Σφ ′) = {f ∈ H∞ (Σφ ×
Σφ ′) : ∃(f1, f2) ∈ H0(Σφ)×H0(Σφ ′), f1 and f2 non-vanishing, and f(f1f2) −1 ∈ H ∞ (Σφ×
Σφ ′)}. Let A, B ∈ S(X) with spectral angles φA and φB, respectively, commute in the
resolvent sense. For φ ∈ (φA, π), φ ′ ∈ (φB, π), and f ∈ H0(Σφ × Σφ
f(A, B) = − 1
4π 2
�
Γψ×Γ ψ ′
′) one defines
:=
f(λ, λ ′ )(λ − A) −1 (λ − B) −1 dλ dλ ′ , (2.12)
where (ψ, ψ ′ ) ∈ (φA, φ) × (φB, φ ′ ). This integral converges in B(X) and does not depend
on the choice of ψ and ψ ′ . Via ΦA, B(f) = f(A, B), it defines a joint functional calculus
ΦA, B : H0(Σφ × Σφ ′) → B(X). In analogy to Definition 2.2.1, we say that (A, B) admits
a bounded joint H∞-calculus (symbolized by (A, B) ∈ H∞ (X)) if there exist φ ∈ (φA, π),
φ ′ ∈ (φB, π), and a constant Kφ, φ ′ < ∞ such that
φ′
|f(A, B)| ≤ Kφ, φ ′|f|φ, ∞ for all f ∈ H0(Σφ × Σφ ′). (2.13)
If (A, B) ∈ H∞ (X), then the functional calculus for (A, B) on H0(Σφ × Σφ ′) extends
uniquely to H∞ (Σφ × Σφ ′).
An interesting question is the following: what are the Banach spaces X for which
(A, B) admits a bounded joint H∞-calculus as soon as A and B, each, admit a bounded
H∞-calculus? In [1] Albrecht was able to prove that this is the case if X = Lp(Ω, Σ, µ),
1 < p < ∞, where (Ω, Σ, µ) is a σ-finite measure space, see also [2, Section 5]. Lancien et
al. [51] extended this result to a class of Banach spaces which enjoy a certain geometric
property. They also give an example for a Banach space not possessing the joint calculus
property. We would further like to mention a result by Kalton and Weis [47] which
asserts, without additional assumption on X, that if A ∈ H∞ (X) and B ∈ RH∞ (X)
with commuting resolvents, then (A, B) admits a bounded joint H∞-calculus. We consider now an important example.
Example 2.4.1 Let 1 < p < ∞, X = Lp(R+ × Rn ), and denote the independent variables
by t (∈ R+) resp. x (∈ Rn ). Take B = ∂t with domain D(B) = 0H 1 p(R+; Lp(Rn )),
and define A as the natural extension of −∆x in Lp(Rn ) with D(−∆x) = H2 p(Rn ) to X,
i.e. D(A) = Lp(R+; H2 p(Rn )) and Af = −∆xf, f ∈ D(A). Then A and B commute in
the resolvent sense, and A, B ∈ H ∞ (X) with H ∞ -angles φ ∞ A = 0 resp. φ∞ B
= π/2. Since
X has the joint calculus property, we have (A, B) ∈ H ∞ (X). More precisely, for each
η ∈ (0, π/2), there exists Cη > 0 such that for all f ∈ H ∞ (Ση × Σ π
2 +η), f(A, B) ∈ B(X)
η, π/2+η
and |f(A, B)| B(X) ≤ Cη|f| ∞ .
Regarding functions in X as elements in Lp(R+; Lp(Rn )), the resolvent of B admits
the kernel representation
(λ − B) −1 w(t) = −
� t
0
e λ(t−s) w(s) ds, t ∈ R+,
for all w ∈ X. Thus, for f ∈ H0(Ση × Σ π
2 +η), the operators f(A, B) admit a kernel
representation as well, namely
� �
t �
−1
f(A, B)w(t) =
e
0 2πi Γψ ′
λ′ �
(t−s) ′ ′
f(A, λ )dλ w(s) ds, t ∈ R+. (2.14)
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