Extension of a regularity result concerning the dam problem (Q1192795)

A question concerning the regularity of the solution of the dam problem is studied: \[ u\in L_ 2(0,T,H^ 1(\Omega)) \quad\text{and}\quad \chi\in L^ \infty(Q); \qquad u\geq 0,\;0\leq\chi\leq 1,\quad u(1- \chi)=0\text{ in }Q;\tag{1} \] \[ u=g \quad\text{on} \quad \Sigma_ D; \qquad \int_ Q (-\chi\partial_ t v+a(\nabla u+\chi e)\cdot\nabla v)\leq 0 \] for every \(v\in H^ 1(Q)\) such that (2) \(v\geq 0\) on \(\Sigma_ D\); \(v=0\) on \(\Sigma_ D\cap\{g>0\}\); \(\nu(\cdot,0)=\nu(\cdot,T)=0\) in \(\Omega\); \(\chi(\cdot,0)=\chi^ 0\) in \(\Omega\), where \(\Omega\) is a connected bounded open set in \(\mathbb{R}^ 2\) wiht Lipschitz boundary (represents the porous medium). The boundary consists of the previous part \(\Gamma_ D\) and the imprevious part \(\Gamma_ N\), whose closures are \(C^{1,1}\) manifolds intersecting a smooth \((n-2)\)-dimensional submanifold of \(\partial\Omega\), \(a=a_{ij}\) is the permeability matrix and \(e\in\mathbb{R}^ n\) is a given unit vector, which takes gravity into account in the physical model; \(Q=\Omega\times ]0,T[\) and \(\Sigma_ D=\Gamma_ D\times]0,T[\). The function \(u\) represents the unknown pressure and the velocity is given by \(-a(\nabla u+\chi e)\) by Darcy's law. The following regularity is assumed for the data: \(g\in C^{0,1}(\mathbb{R}^{n+1})\), \(g\geq 0\), \(\chi^ 0\in L^ \infty(\Omega)\), \(0\leq\chi^ 0\leq 1\). Main result: One proves, in the case of piecewise smooth coefficients, that the time derivative of the solution of the dam problem (1),(2) is a measure, extending the result proved by the same authors in the case of Lipschitz continuous coefficients.