Quasilinear parabolic problems with nonlinear boundary conditions

We give now an overview of the contents of the thesis and present the principal
ideas in greater detail. The text is divided into three main parts, devoted respectively
to preliminaries (Chapter 2), linear theory (Chapters 3, 4, 5), and nonlinear problems
(Chapter 6).
Chapter 2 collects the basic tools needed for the investigation of the linear equations
to be studied. After fixing some notations, in Section 2.2 we review important classes of
sectorial operators, among others, operators which admit a bounded H∞-calculus, operators
with bounded imaginary powers, and R-sectorial operators. We further discuss
some properties of the fractional powers of such operators in connection with real and
complex interpolation, and prove that the power Aα , α ∈ R, of an R-sectorial operator
A with R-angle φR A is R-sectorial, too, as long as the inequality |α|φR A < π holds (Propo-
sition 2.2.1); the latter result seems to be missing in the literature. In Section 2.3, which
is devoted to sums of closed linear operators, we state a variant of the Dore-Venni theorem.
Section 2.4 is concerned with the joint H ∞ -calculus for pairs of sectorial operators.
In particular, we look at the calculus for the pair (∂t, −∆x) in the space Lp(R+ × R n ),
which proves extremely useful in establishing optimal regularity results in Chapter 5.
Section 2.5 deals with operator-valued Fourier multipliers. The central result here is the
Mikhlin multiplier theorem in the operator-valued version, which was proven recently by
Weis [80]. In Section 2.6 we introduce the class of K-kernels consisting of all 1-regular,
sectorial kernels k whose Laplace transform ˆ k(λ) behaves like λ −α as λ → 0, ∞ for some
α ≥ 0; an example is given by k(t) = t α−1 e −βt , t ≥ 0, with α > 0 and β ≥ 0. Kernels of
that type have already been studied by Prüss [63] in the context of Volterra operators
in Lp, which is the subject of Section 2.8. Before, in Section 2.7 we give a short account
of the abstract Volterra equation
u(t) +
� t
0
a(t − s)Au(s) ds = f(t), t ≥ 0, (1.16)
where a ∈ L1, loc(R+) is a scalar kernel, and A is a closed linear operator in a Banach
space X. We explain the notion of parabolicity of (1.16), give the definition of resolvents,
and recall the variation of constants formula. Section 2.8 is devoted to convolution
operators in Lp associated to a K-kernel. After stating two fundamental theorems from
Prüss [63] on the inversion of such operators in Lp(R; X) with 1 < p < ∞ and X ∈ HT ,
we consider restrictions of them to Lp(J; X), where J = [0, T ] or R+. The main facts
about these operators are summarized in Corollary 2.8.1. It asserts that for every Kkernel
k with angle θk < π there is a unique sectorial operator B in Lp(J; X) inverting
the convolution (k∗), and that this operator - assuming in addition k ∈ L1(R+) in case
J = R+ - is invertible and satisfies B −1 w = k ∗ w for all w ∈ Lp(J; X); it further
says that B ∈ BIP(Lp(J; X)) and that its domain D(B) equals the space 0H α p (J; X),
where α ≥ 0 refers to the order of k in the sense describe above. So, we have precise
information about the mapping properties of the convolution operators under study and
see that their inverse operators are accessible to the Dore-Venni theorem. In Section
2.8 we further recognize the fractional derivative (d/dt) α of order α ∈ (0, 1) to be the
inverse convolution operator associated with the standard kernel t α−1 /Γ(α). Besides, we
introduce equivalent norms for the spaces H α p (J; X) and consider operators of the form
(I − k∗), which appear in connection with transformations of Volterra equations, cf. the
above example on heat conduction.
The main purpose of Chapter 3 is to establish maximal regularity results of type
Lp for equation (1.16) as well as for a class of abstract linear Volterra equations on an
infinite strip J×R+ with inhomogeneous boundary condition of Dirichlet resp. (abstract)
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