Degenerate elliptic operators with boundary conditions via form methods

Abstract

The main subject in the first part of this thesis are form methods. Abstractly, form methods provide a means of both defining and studying unbounded operators in a Hilbert space. The probably most well-known instance of a form method is Kato's representation theorem for closed sectorial forms (1966). This result is commonly applied to obtain suitable realisations of elliptic differential operators in divergence form as unbounded operators in L²-space. Form methods tend to be quite robust, which is particularly useful for perturbation problems. Recently, Arendt and ter Elst (2008) have extended Kato's representation theorem to general sectorial forms, without the closedness condition and relaxing the former requirement that the form domain is embedded in the Hilbert space. This extension is well-suited for the degenerate elliptic setting and has also been applied to the Dirichlet-to-Neumann operator. The main contributions of this thesis regarding form methods are the introduction of an abstract form method for accretive forms, the study of compactly elliptic forms including an application to the convergence of generalised Dirichlet-to-Neumann graphs and an investigation of the regular part of sectorial forms providing a formula for the important case of second-order differential sectorial forms. This part of the thesis includes joint work with Wolfgang Arendt, Tom ter Elst, James Kennedy and Hendrik Vogt. In the final chapter of the thesis we consider a notion of a weak trace for elements of a Sobolev space, which is related to work of Maz'ya and arose in the study of the Laplacian with Robin boundary conditions and the Dirichlet-to-Neumann operator on arbitrary domains. Using tools from potential and lattice theory, we investigate the space of elements with weak trace zero. This is related to questions regarding the stability of the Dirichlet problem for varying domains.