On the shape of the free boundary of variational inequalities with gradient constraints (Q2364614)

Summary: In this article we derive an estimate on the number of local maxima of the free boundary of the minimizer of \[ I[v]:=\int_U\frac12|Dv|^2-\eta v\,dx, \] subject to the pointwise gradient constraint \[ (|D_1v|^q+|D_2v|^q)^{\frac1q}\leqslant1. \] This also gives an estimate on the number of connected components of the free boundary. In addition, we further study the free boundary when \(U\) is a polygon with some symmetry.