arXiv subjects
Publications and source records attributed to Christoph Scheven.
We study generalized products of divergence-measure fields and gradient measures of {\rm BV} functions. These products depend on the choice of a representative of the {\rm BV} function, and here we single out a specific choice which is suitable in order to define and investigate a notion of weak supersolutions for the $1$-Laplace operator.
We introduce higher order variants of the Yang-Mills functional that involve $(n-2)$th order derivatives of the curvature. We prove coercivity and smoothness of critical points in Uhlenbeck gauge in dimensions $\mathrm{dim}M\le 2n$. These results are then used to establish the existence of smooth minimizers on a given principal bundle $P\to M$ for subcritical dimensions $\mathrm{dim}M<2n$. In the case of critical dimension $\mathrm{dim}M=2n$ we construct a minimizer on a bundle which might differ from the prescribed one, but has the same Chern classes $c_1,\ldots,c_{n-1}$. A key result is a removable singularity theorem for bundles carrying a $W^{n-1,2}$-connection. This generalizes a recent result by Petrache and Rivière.
In this paper we consider the heat flow associated to the classical Plateau problem for surfaces of prescribed mean curvature. We show that an isoperimetric condition on H ensures the existence of a global weak solution. Moreover, we establish that these global solutions sub-converge, as time tends to infinity, to a conformal solution of the classical Plateau problem for surfaces of prescribed mean curvature.