\(C^{1,1}\)-regularity of constrained area minimizing hypersurfaces (Q1181696)

In a previous paper [``The constrained least gradient problem in \(R^ n\)'' (to appear in Trans. Am. Math. Soc.)], the author considered the constrained least gradient problem \[ \inf\left\{\int_ \Omega| \nabla u| dx:\;u\in C^{0,1}(\bar\Omega),\;| \nabla u| \leq 1\hbox{ a.e., }u=g\hbox{ on }\partial\Omega\right\} \] where \(\Omega\subset R^ n\) is an open set and \(g\) is a Lipschitz function on \(\partial \Omega\). The solution \(u\) was constructed by equating the set \(\{u\geq t\}\) with the solution to the following obstacle problem: \[ \inf\{P(E,\Omega):\;\bar\Omega\supset E\supset L,\;\bar E\cap (M)^ i=\emptyset\} \] where \(P(E,\Omega)\) denotes the perimeter of \(E\) in \(\Omega\) and \((M)^ i\) denotes the topological interior of \(M\). \(L\) and \(M\) are closed sets that depend on \(t\) and satisfy an interior ball condition of radius \(R\). This paper investigates the regularity of the solution to the obstacle problem. The strongest previous result on this question was due to \textit{I. Tamanini} [J. Reine Angew. Math. 334, 27-39 (1982; Zbl 0479.49028)] who showed that a minimizer \(E\) satisfies \(\partial E\in C^{1,1/2}\) in a neighborhood of \(\partial L\cup \partial M\). It is shown that Tamanini's result can be improved to \(C^{1,1}\) regularity and that this regularity is essentially optimal.