Existence of minima for nonconvex functionals in spaces of functions depending on the distance from the boundary (Q1968729)

The author studies the following problem \[ \underset{\mathcal H} {\text{Min}} \int_\Omega [h(\nabla u)+ u] dx, \] where \(\Omega\) is an open bounded subset of \(\mathbb{R}^2\), \(h\) is not necessarily a convex function and \({\mathcal H}\) is the subset of \(W^{1,1}_0\) of the functions depending only on the distance from the boundary \(\partial\Omega\). When \(\Omega\) is a disk centered at the origin these are the radially symmetric functions and the problem has been solved by Cellina and Perrotta. Here the author considers the case where \(\Omega\) is a regular polygon \(\Omega_n\) of \(n\) sides inscribed in the disk and proves that there is only a solution \(u_n\). Moreover, he proves that the sequence \(\{u_n\}\) converges uniformly to the unique solution \(u\) in the disk for \(n\to+\infty\).