Wentzell boundary conditions for elliptic fourth-order operators

Abstract

Wentzell (or dynamic) boundary conditions model the interchange of free energy between the boundary and the interior of a physical system, which classical boundary conditions like Dirichlet or Neumann conditions cannot capture. They pose a mathematically challenging problem as the differential operator driving the evolution equation in the interior is also allowed to appear on the boundary, whence existence of the appearing traces is a priori unclear. This thesis deals mostly with elliptic operators of order four where classical methods like Beurling-Deny criteria cannot be used as the operator is not positive. In order to overcome these issues, two main approaches are discussed in this thesis:

In the first part, form methods and weak abstract traces are used in the Hilbert space setting in order to show that the associated operator in the product space L2 x L2 (where the parts of the functions on interior and boundary are decoupled) generates a real analytic semigroup of contractions, even though the domain only possesses Lipschitz regularity. Furthermore, the solution is shown to be Hölder-continuous, its long-time behavior is investigated, and criteria for stability and eventual positivity are given.

In the second part, in the Banach space setting with smoother boundary, classical parabolic methods are used, which leads to a more general class of boundary value problems with rough data as the problem is not investigated in the classical trace spaces but in Lp x Lp. By introducing a special type of anisotropic Sobolev spaces, which are compatible with interpolation and also allow a good characterization of multipliers, those boundary value problems are solved in the half-space, and by embeddings into classical spaces and localization a solution theory is obtained on domains, as well. Finally, those results are applied to the Wentzell case, which (on the half space) also leads to an analytic semigroup.