BV estimates on the transport density with Dirichlet region on the boundary (Q6574278)

The paper under review studies the regularity of the transport density \(\sigma\) in a PDE system of Monge-Kantorovich type, namely Equation (1.4) therein: \N\[\N\begin{cases} \N-\nabla \cdot [\sigma \nabla u] = f \qquad & \text{ in } \mathring{\Omega},\\\Nu = g \qquad & \text{ on } \partial \Omega,\\\N|\nabla u | \leq 1 \qquad & \text{ in } \Omega,\\\N|\nabla u| = 1\qquad & \sigma\text{-a.e.}\N\end{cases}\N\]\NHere \(\Omega\) is a compact domain \(\mathbb{R}^2\) with \(C^{2,1}\)-boundary. The Dirichlet boundary data \(g\) in in \(C^{2,1}(\partial \Omega)\) and \(\beta\)-Lipschitz in \(\Omega\) with \(\beta<1\).\N\NIt is proved that if \(f\) is in \(BV(\Omega)\cap L^\infty(\Omega)\) (or \(W^{1,1}(\Omega)\cap L^\infty(\Omega)\), resp.), then \(\sigma\) is of BV-regularity (or \(W^{1,1}\)-regularity, resp.). A counterexample is constructed that \(f \in C^\infty\) does \textit{not} imply \(\sigma \in W^{1,5}\).\N\NThe techniques used in this paper are specific for two dimensions. The BV-regularity in higher dimensions remains open.