  In this paper local Lipschitz regularity of weak solutions to certain
singular elliptic equations involving one-Laplacian is studied. Equations
treated here also contains another well-behaving elliptic operator such as
$p$-Laplacian with $1<p<\infty$. The problem is that one-Laplacian is too
singular on degenerate points, what is often called facet, which makes it
difficult to obtain even Lipschitz regularity of weak solutions. This
difficulty is overcome by making suitable approximation schemes, and by
avoiding analysis on facet for approximated solutions. The key estimate is a
local a priori uniform Lipschitz estimate for classical solutions to
regularized equations, which is proved by Moser's iteration. Another local a
priori uniform Lipschitz bounds can also be obtained by De Giorgi's truncation.
Proofs of local Lipschitz estimates in this paper are rather classical and
elementary in the sense that nonlinear potential estimates are not used at all.
