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

Mohammad Safdari

doi:10.4171/ifb/380

JournalsifbVol. 19, No. 2pp. 183–200

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

Mohammad Safdari
Institute for Research in Fundamental Sciences (IPM), Tehran, Iran

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Abstract

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} ∣ D v ∣^{2} - η v d x,

subject to the pointwise gradient constraint

(∣ D_{1} v ∣^{q} + ∣ D_{2} v ∣^{q})^{\frac{1}{q}} \leq 1.

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.

Cite this article

Mohammad Safdari, On the shape of the free boundary of variational inequalities with gradient constraints. Interfaces Free Bound. 19 (2017), no. 2, pp. 183–200

DOI 10.4171/IFB/380