Regularity of stationary surfaces of Cartan functionals with Plateau boundary conditions

Haeger, Henrik; von der Mosel, Heiko (Thesis advisor); Wagner, Alfred (Thesis advisor)

Aachen (2020)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2020

Abstract

The thesis discusses the regularity of stationary surfaces of Cartan functionals with Plateau boundary conditions. These can be seen as a generalization of minimizers of two-dimensional parametric functional streated in [HvdM03b] and [HvdM03c]. All results are derived from a weak form of an Euler-Lagrange inequality with respect to test functions which preserve the boundary constraints. We make use of the concept of dominance functions for Cartan integrands (see [HvdM03a])which provides an Euler-Lagrange inequality of a "better" quadratic functional. From that, we can formulate several conditions under which Hölder continuity can be achieved, which is the starting point for other regularity results. In order to obtain higher regularity, the existence of a perfect dominance function is required. Moreover, we establish smoothness of stationary surfaces away from its branching points. The final result raises the power of integrability of the hessian of a surface (guaranteed its existence) using that the gradient minimizes an elliptic functional constructed by Kipps [Kip57].