Obstacle problems with linear growth: Hölder regularity for the dual solutions (Q2774472)

The authors consider obstacle problems of the form NEWLINE\[NEWLINE\text{Minimize }J(u)=\int_\Omega f(\nabla u) dx,NEWLINE\]NEWLINE where \(\Omega\) is a bounded domain of \(R^n\), \(u\) is a function from \(\Omega \) into \({R}\), \(u=0\) on \(\partial \Omega\), \(u\geq \psi\), \(\psi\) is a given smooth function such that \(\psi< 0\) on \(\partial \Omega\). The function \(f\) is smooth, convex, and it satisfies linear growth conditions. Due to the linear growth condition on \(f\) the problem does not necessarily admit solutions in \(W^{1,1}_0(\Omega)\). The authors study the dual problem. They prove that the dual problem admits a unique solution \(\sigma\), this solution belongs to \(W^{1,2}_{\text{loc}}(\Omega;R^n)\), and \(\sigma\) is Hölder continuous in the interior of \(\Omega\) with any exponent \(\alpha \in (0,1)\). Introducing \(\widehat J\), the lower semicontinuous envelope of \(J\) with respect to the \(L^1_{\text{loc}}\)-convergence, the proof of the regularity of \(\sigma\) is based on the analysis of the regularity of a local minimizer \(u^*\) for \(\widehat J\).