  We consider a Neumann problem for strictly convex variational functionals of
linear growth. We establish the existence of minimisers among
$\operatorname{W}^{1,1}$-functions provided that the domain under consideration
is simply connected. Hence, in this situation, the relaxation of the functional
to the space of functions of bounded variation, which has better compactness
properties, is not necessary. Similar $\operatorname{W}^{1,1}$-regularity
results for the corresponding Dirichlet problem are only known under rather
restrictive convexity assumptions limiting its non-uniformity up to the
borderline case of the minimal surface functional, whereas for the Neumann
problem no such quantified version of strong convexity is required.
