Gradient estimate in terms of a Hilbert-like distance,for minimal surfaces and Chaplygin gas

We consider a quasilinear elliptic boundary value-problem with homogenenous Dirichlet condition. The data are a convex planar domain. The gradient estimate is needed to ensure the uniform ellipticity, before applying regularity theory. We establish this estimate in terms of a distance, which is equivalent to the Hilbert metric.

This fills the proof of existence and uniqueness of a solution to this BVP (boundary-value problem), when the domain is only convex but not strictly, for instance if it is a polygon.