Partial regularity for evolution problems with discontinuity (Q1914851)

For a bounded Lipschitz domain \(\Omega\) in \(\mathbb{R}^n\) and a \(C^1\) convex linear-growth function \(\phi(p)\) on \(\mathbb{R}^n\), the first author and Zhou studied the existence and uniqueness questions on the evolution problem \[ {\partial u\over \partial t}= \sum^n_{i= 1} {\partial\over \partial x_i} \Biggl[ {\partial\phi\over \partial p_i} (\nabla u)\Biggr], \] with suitable Dirichlet boundary data in [the first author and \textit{X. Zhou}, Commun. Partial Differ. Equations 19, 1879-1907 (1994; Zbl 0811.35061)]. The unique weak solutions belong to the function space \(L^2([0, \infty]: \text{BV}(\Omega))\), so that they may in general have discontinuities in the space variable at each time. In fact, depending on the function \(\phi\), the initial and boundary data, some weak solutions do have discontinuities as one can demonstrate with simple examples. In this paper, we establish certain partial regularity results for such solutions in case the space dimension is equal to one or two, with a necessary local coerciveness assumption on \(\phi\).