  This work is about global H\"older regularity for solutions to elliptic
partial differential equations subject to mixed boundary conditions on
irregular domains. There are two main results. In the first, we show that if
the domain of the realization of an elliptic differential operator in a
negative Sobolev space with integrability $q > d$ embeds into a space of
H\"older continuous functions, then so do the domains of suitable fractional
powers of this operator. The second main result then establishes that the
premise of the first is indeed satisfied. The proof goes along the classical
techniques of localization, transformation and reflection which allows to fall
back to the classical results of Ladyzhenskaya or Kinderlehrer. One of the main
features of our approach is that we do not require Lipschitz charts for the
Dirichlet boundary part, but only an intriguing metric/measure-theoretic
condition on the interface of Dirichlet- and Neumann boundary parts. A similar
condition was posed in a related work by ter Elst and Rehberg in 2015, but the
present proof is much simpler, if only restricted to space dimension up to 4.
