Quasilinear parabolic problems with nonlinear boundary conditions

Let us define the sum operator A + B by
(A + B)x = Ax + Bx, x ∈ D(A + B) = D(A) ∩ D(B).
If 0 ∈ ρ(A+B), which in particular means that A+B is closed, equation (2.9) is solvable
in the strict sense for all y ∈ X, and the closed graph theorem shows (2.10) with some
C > 0. For the latter it suffices to know only that A+B is injective and closed. If A+B
is merely closable but not closed, and 0 ∈ ρ(A + B), then (2.9) only admits generalized
solutions in the sense that there exist sequences (xn) ⊂ D(A) ∩ D(B), xn → x, and
yn → y satisfying
Axn + Bxn = yn, n ∈ N.
In general, nothing can be said on A + B. It may even happen that it is not closable.
In order to prove positive results in this direction, further assumptions on A and B have
to be imposed.
In 1975 Da Prato and Grisvard ([26]) were able to show that if A and B are commuting
sectorial operators satisfying the parabolicity condition φA + φB < π then A + B
is closable, and the closure L := A + B is a sectorial operator with φL ≤ max{φA, φB},
see also [16], [63, Section 8]. Recall that two closed linear operators are said to commute
(in the resolvent sense) if there are λ ∈ ρ(A), µ ∈ ρ(B) such that
(λ − A) −1 (µ − A) −1 = (µ − A) −1 (λ − A) −1 .
By strengthening the assumptions on A, B and X Dore and Venni [31], [32] succeeded
in proving closedness of A + B. Prüss and Sohr [69] improved their result by removing
some extra assumptions. Before we repeat a version of the Dore-Venni theorem we have
to recall what it means for a Banach space X to belong to the class HT .
A Banach space X is said to be of class HT , if the Hilbert transform is bounded
on Lp(R, X) for some (and then all) p ∈ (1, ∞). Here the Hilbert transform Hf of a
function f ∈ S(R; X), the Schwartz space of rapidly decreasing X-valued functions, is
defined by
(Hf)(t) = 1
π lim
�
ɛ→0 ɛ ≤|s|≤ 1/ɛ
f(t − s) ds
, t ∈ R,
s
where the limit is to be understood in the Lp-sense. There is a well known theorem
which says that the set of Banach spaces of class HT coincides with the class of UMD
spaces, where UMD stands for unconditional martingale difference property. It is further
known that HT -spaces are reflexive. Every Hilbert space belongs to the class HT , and
if (Ω, Σ, µ) is a measure space, 1 < p < ∞, then Lp(Ω, Σ, µ; X) is an HT -space. For all
these results see the survey article by Burkholder [10].
We state now a variant of the Dore-Venni theorem, cf. [31], [65], [69].
Theorem 2.3.1 Suppose X is a Banach space of class HT , and assume A, B ∈ BIP(X)
commute in the resolvent sense and satisfy the strong parabolicity condition θA+θB < π.
Let further µ > 0. Then
(i) A + µB is closed and sectorial;
(ii) A + µB ∈ BIP(X) with θA+µB ≤ max{θA, θB};
(iii) there exists a constant C > 0, independent of µ > 0, such that
|Ax| + µ|Bx| ≤ C|Ax + µBx|, x ∈ D(A) ∩ D(B). (2.11)
In particular, if A or B is invertible, then A + µB is invertible as well.
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