Boundary value problems with rough boundary data

Prof. Dr. Robert Denk, Universität Konstanz

A typical example of a boundary value problem is given by
(λ − ∆)u = f in G,
u = g on ∂G,

where λ > 0 and G ⊂ R^n is a bounded and sufficiently smooth domain. For f ∈ L²(G), the natural solution space for u is the second-order Sobolev space H²(G). For the boundary data g, the canonical space is given by H^(3/2)(∂G) – this is a Sobolev space of non-integer order, which indicates a connection to the theory of function spaces. If f ∈ L^p(G) for some p ∈ (1, ∞), we get u ∈ W^2_p(G), and the canonical boundary space is the Besov space of order 2 − 1/p.

In some applications, the boundary data are not smooth enough to apply the classical theory. This happens, for instance, if we have a stochastic force term on the boundary (boundary noise) and/or some dynamics on the boundary. For this, one has to generalize the trace map u → u|∂G, including even Besov spaces of negative order for the boundary data. One can show unique solvability for a general class of boundary value problems and the generation of an analytic semigroup in the case of dynamic boundary conditions. Applications include the Bi-Laplacian with Wentzell boundary conditions, the linearized Cahn-Hilliard equation with dynamic boundary conditions, and coupled plate-membrane systems.