Generalized solutions for a class of non-uniformly elliptic equations in divergence form

We study a general class of quasilinear non-uniformly elliptic pdes in divergence form with linear growth in the gradient. We examine the notions of BV and viscosity solutions and derive for such generalized solutions various a priori pointwise and integral estimates, including a Harnack inequality. In particular we prove that viscosity solutions are unique (on strictly convex domains), are contained in the space BVloc and are C-1,C-alpha almost everywhere.