In this article we derive an estimate on the number of local maxima of the free boundary of the minimizer of

$$
I[v]:=\int_{U}\frac{1}{2}|Dv|^{2}-\eta v\,dx,
$$

subject to the pointwise gradient constraint

$$
(|D_{1}v|^{q}+|D_{2}v|^{q})^{\frac{1}{q}}\le1.
$$

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.

On the shape of the free boundary of variational inequalities with gradient constraints

Mohammad Safdari

Abstract

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