Second order abstract initial - boundary value problems

Introduction Partial differential equations on bounded domains of R n have traditionally been equipped with homogeneous boundary conditions (usually Dirichlet, Neumann, or Robin). However, other kinds of boundary conditions can also be considered, and for a number of concrete application it seems that dynamic (i.e., time-dependent) boundary conditions are the right ones. Motivated by physical problems, numerous partial differential equations with dynamic boundary conditions have been studied in the last decades: H.cher has investigated parabolic problems (see [Es93] and references therein); and J.T. Beale and V.N. Krasil'nikov, among others, have investigated second order hyperbolic equations with dynamical boundary conditions (see [Be76], [Kr61], [Be00], and references therein). In recent years, a systematic study of problems of this kind has been performed mainly who in a series of papers (see [FGGR02], [FGG+03], and references therein) have convincingly shown that dynamic boundary conditions are the natural L p-counterpart to the well-known (generalized) Wentzell boundary conditions. On the other side, K.-J. Engel has introduced a powerful abstract technique to handle this kind of problems, reducing them in some sense to usual, perturbed evolution equations with homogeneous, time-independent boundary conditions (see [En99], [CENN03], and [KMN03]). Both schools reduce the problem to an abstract Cauchy problem associated to an operator matrix on a suitable product space. We remark that more recently an abstract approach that in some sense unifies dynamic and static boundary value problems has been developed by G. Nickel, cf. [Ni04]. In the first chapter we introduce an abstract setting to consider what we call an abstract initial boundary value problem, i.e., a system of the form (AIBVP)            ˙ u(t) = Au(t), t ≥ 0, ˙ x(t) = Bu(t) + ˜ Bx(t), t ≥ 0, x(t) = Lu(t), t ≥ 0, u(0) = f ∈ X, x(0) = g ∈ ∂X. Here the first equation takes place on a Banach state space X (in concrete applications, this is often a space of functions on a domain Ω ⊂ R n with smooth, 1 nonempty boundary ∂Ω). The third equation represents a coupling relation between the variable in X and the variable in a Banach boundary space ∂X (in concrete applications, this is often a space of functions on ∂Ω). Finally, the second equation represents an evolution equation on the boundary with a feedback term given by the operator B. Following [KMN03, § 2], …