Quasilinear parabolic problems with nonlinear boundary conditions

Chapter 4
Linear Problems of Second Order
This chapter is devoted to the study of linear problems of second order on Lp(J × Ω),
Ω a domain in R n , with general inhomogeneous boundary conditions of order ≤ 1. We
shall apply abstract results proven in Chapter 3 to characterize maximal Lp-regularity
of the solutions in terms of regularity and compatibility conditions for the data. We will
first consider problems on the full space R n . This will be followed by the investigation
of the half space case. Finally, we study the case of an arbitrary domain. Here we use
the localization method to reduce the problem to related problems on R n and R n +.
4.1 Full space problems
In this section we study the Volterra equation
v + k ∗ A(·, x, Dx)v = f, t ∈ J, x ∈ R n , (4.1)
in Lp(J; Lp(R n )). Here J = [0, T ], the kernel k belongs to the class K 1 (α, θ) with
α ∈ (0, 2), θ < π, and A(t, x, Dx) is a differential operator of second order with variable
coefficients:
A(t, x, Dx) = −a(t, x) : ∇ 2 x + b1(t, x) · ∇x + b0(t, x), t ∈ J, x ∈ R n . (4.2)
By ∇2 xv we mean the Hessian matrix of v w.r.t. x, that is (∇2 xv(t, x))ij = ∂xi∂xj v(t, x),
i, j = 1, . . . , n. The double scalar product a : b of two matrices a, b ∈ Cn×n is defined
by a : b = � n
i, j=1 aijbij. We further denote the principal part of the differential operator
(4.2) by A#(t, x, Dx), that is A#(t, x, Dx) = −a(t, x) : ∇ 2 x, and we write A(t, x, Dx) =
A#(t, x, Dx) + AR(t, x, Dx).
For an unbounded domain Ω ⊂ R n and a Banach space X, we set
Cul(J × Ω; X) = {g ∈ C(J × Ω; X) : lim g(t, x) exists uniformly for all t ∈ J}.
|x|→∞
The symbol Sym{n} stands for the space of n-dimensional real symmetric matrices.
The goal of this section is to prove the following result.
Theorem 4.1.1 Let 1 < p < ∞ and n ∈ N. Suppose the differential operator A(t, x, Dx)
is given by (4.2). Assume the following properties.
(H1) k ∈ K 1 (α, θ), where α ∈ (0, 2) \ {1/p, 1 + 1/p}, θ < π;
57