Quasilinear parabolic problems with nonlinear boundary conditions

conditions of order ≤ 1. Sections 4.1 and 4.2 deal with the full resp. half space case.
The theory from Chapter 3 is applied to find necessary and sufficient conditions on the
data that characterize maximal Lp-regularity of the solutions. The strategy is to look
first at problems with constant coefficients and differential operators consisting only of
their principle parts, and then to use pointwise multiplication properties of the function
spaces involved together with perturbation arguments to extend the results to the general
case. In Section 4.3 we study the case of an arbitrary domain Ω ⊂ R n with compact
C 2 -smooth boundary Γ decomposing into two disjoint closed parts ΓD and ΓN on which
inhomogeneous boundary conditions of zeroth resp. first order have to be satisfied. The
basic idea here is to employ the localization method to reduce the problem to related
problems on R n and R n +. Proceeding this way we obtain a characterization of unique
solvability of the problem
with
⎧
⎨
⎩
v + k ∗ A(·, x, Dx)v = f, t ∈ J, x ∈ Ω,
v = g, t ∈ J, x ∈ ΓD,
B(t, x, Dx)v = h, t ∈ J, x ∈ ΓN,
A(t, x, Dx) = −a(t, x) : ∇ 2 x + a1(t, x) · ∇x + a0(t, x), t ∈ J, x ∈ Ω,
B(t, x, Dx) = b(t, x) · ∇x + b0(t, x), t ∈ J, x ∈ ΓN,
(1.19)
in the regularity class H α p (J; Lp(Ω)) ∩ Lp(J; H 2 p(Ω)). Here as before, k is a K-kernel
of order α ∈ (0, 2). As to the regularity of the top order coefficients, we only assume
a ∈ C(J × Ω, R n×n ) and that a has a limit as |x| → ∞ uniformly w.r.t. t ∈ J.
In Chapter 5 we are concerned with a linear parabolic problem of second order which
appears in the theory of viscoelasticity. In comparison to the problems investigated
in Chapter 4, it has two new challenging features: first it is a vector-valued problem,
and second it contains two independent kernels. Once more we characterize unique
existence of the solution in a certain class of optimal regularity in terms of regularity
and compatibility conditions on the given data. Section 5.1 gives a short account of the
basic equations of linear viscoelasticity. In Section 5.2 we state the problem and discuss
the assumptions on the kernels, which are stronger than in the previous chapters, since
the method of proof relies heavily on H ∞ -calculus. Section 5.3 is devoted to the thorough
investigation of a half space case of the problem under study, which reads
⎧
⎪⎨
⎪⎩
∂tv − da ∗ (∆xv + ∂2 yv) − (db + 1
∂tw − da ∗ ∆xw − (db + 4
3da) ∗ ∂2 yw − (db + 1
3da) ∗ (∇x∇x · v + ∂y∇xw) = fv (J × R n+1
+ )
3da) ∗ ∂y∇x · v = fw (J × R n+1
+ )
−da ∗ γ∂yv − da ∗ γ∇xw = gv (J × Rn )
−(db − 2
3 da) ∗ γ∇x · v − (db + 4
3 da) ∗ γ∂yw = gw (J × R n )
v|t=0 = v0 (R n+1
+ )
w|t=0 = w0 (R n+1
+ ),
where the unknown functions v and w are R n - resp. scalar-valued, and γ denotes the
trace operator at y = 0. To solve this problem, we introduce an appropriate auxiliary
function by which the system can be decoupled. Using the results from Chapter 3, the
problem is further reduced to an equation on the boundary, which can be solved by
means of the joint H ∞ -calculus for the pair (∂t, −∆x) in the space Lp(R+ × R n ). The
essential difficulty is the estimate for the principal symbol of the problem. To be precise,
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