  In this paper, we consider a convex function defined as a 1D-regularized
total variation with nonhomogeneous coefficients, and prove the Main Theorem
concerned with the decomposition of the subdifferential of this convex function
to a weighted singular diffusion and a linear regular diffusion. The Main
Theorem will be to enhance the previous regularity result for quasilinear
equation with singularity, and moreover, it will be to provide some useful
information in the advanced mathematical studies of grain boundary motion,
based on KWC type energy.
