Quasilinear parabolic problems with nonlinear boundary conditions

Chapter 2
Preliminaries
2.1 Some notation, function spaces, Laplace transform
In this section we fix some of the notations used throughout this thesis, recall some basic
definitions and give references concerning function spaces and the Laplace transform.
By N, Z, R, C we denote the sets of natural numbers, integers, real and complex
numbers, respectively. Let further N0 = N∪{0}, R+ = [0, ∞), C+ = {λ ∈ C : Re λ > 0}.
X, Y, Z will usually be Banach spaces; | · |X designates the norm of the Banach space
X. The symbol B(X, Y ) means the space of all bounded linear operators from X to Y ,
we write B(X) = B(X, X) for short. If A is a linear operator in X, D(A), R(A), N (A)
stand for domain, range, and null space of A, respectively, while ρ(A), σ(A) designate
resolvent set and spectrum of A. For a closed operator A we denote by DA the domain
of A equipped with the graph norm.
In what follows let X be a Banach space. For Ω ⊂ Rn open or closed, C(Ω; X)
and BUC(Ω; X) stand for the continuous resp. bounded uniformly continuous functions
f : Ω → X. Further, if Ω ⊂ Rn is open and k ∈ N, Ck (Ω; X) (BUC k (Ω; X)) designates
the space of all functions f : Ω → X for which the partial derivative ∂αf exists on Ω
and can be continuously extended to a function belonging to C(Ω; X) (BUC(Ω; X)), for
each 0 ≤ |α| ≤ k.
If Ω is a Lebesgue measurable subset of Rn and 1 ≤ p < ∞, then Lp(Ω; X) denotes
the space of all (equivalence classes of) Bochner-measurable functions f : Ω → X with
|f|p := ( �
Ω |f(y)|p X dy)1/p < ∞. Lp(Ω; X) is a Banach space when normed by | · |p.
For an interval J ⊂ R, s > 0 and 1 < p < ∞, by Hs p(J; X) and Bs pp(J; X) we
mean the vector-valued Bessel potential space resp. Sobolev-Slobodeckij space of Xvalued
functions on J, see Amann [6], Schmeisser [73], ˇ Strkalj [76], and Zimmermann
[83]. Concerning the scalar case, we refer further to Runst and Sickel [72], Triebel [78],
[79]. In Section 2.8 we give a definition of the spaces Hs p(J; X) in the situation where
X belongs to the class HT (cf. Section 2.3); this will always be the case when we are to
consider vector-valued Bessel potential spaces. We will frequently use the property that
the Sobolev-Slobodeckij spaces appear as real interpolation spaces between the spaces
Lp and Hs p; more precisely (Lp(J; X), Hs p(J; X))θ, p = Bθs pp(J; X) for all 1 < p < ∞,
s > 0, and θ ∈ (0, 1). For general treatises on interpolation theory we refer to Bergh
and Löfström [9], and Triebel [78].
If F is any of the above function spaces, then f ∈ Floc means that f belongs to the
corresponding space when restricted to compact subsets of its domain. In the scalar
case X = R or X = C we usually omit the image space in the function space notation.
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