  We study the minimization of convex, variational integrals of linear growth
among all functions in the Sobolev space $W^{1,1}$ with prescribed boundary
values (or its equivalent formulation as a boundary value problem for a
degenerately elliptic Euler--Lagrange equation). Due to insufficient
compactness properties of these Dirichlet classes, the existence of solutions
does not follow in a standard way by the direct method in the calculus of
variations and in fact might fail, as it is well-known already for the
non-parametric minimal surface problem. Assuming radial structure, we establish
a necessary and sufficient condition on the integrand such that the Dirichlet
problem is in general solvable, in the sense that a Lipschitz solution exists
for any regular domain and all prescribed regular boundary values, via the
construction of appropriate barrier functions in the tradition of Serrin's
paper [19].
