On the Hausdorff dimension and singularities of the monopolist's free boundary curve

TL;DR

This paper investigates the geometric and regularity properties of the free boundary in a multidimensional monopolist's problem, establishing its Hausdorff dimension, regularity, and singularity structure.

Contribution

It proves that the free boundary is a Hausdorff dimension one curve with $C^eta$ regularity, except at a discrete set of singular points, advancing understanding of free boundary regularity in this context.

Findings

The free boundary has Hausdorff dimension one.

The free boundary is $C^eta$ for all $0<eta<1$ outside a discrete set.

The free boundary becomes $C^ abla$ outside a closed set with empty relative interior.

Abstract

The simplest genuinely multidimensional monopolist's problem involves minimizing a linearly perturbed Dirichlet energy among nonnegative convex functions $u$ on an open domain $X \subset [0, \infty)^2$. The geometry of the region of strict convexity $\Omega\subset X$ for the unique minimizer $u$ is of central interest. A relatively closed portion $X_1^0 \subset X$ of the domain is comprised of line segments starting and ending on $\partial X$ along which $u$ is affine. For convex polygons and potentially all domains $X \subset \mathbf{R}^2$, we build on results with Zhang to show that outside $X_1^0 \cup \{u=0\}$, the free boundary of $\Omega$ is a continuous curve of Hausdorff dimension one, and that $\Omega$ has density $1/2$ along it (and is $C^\alpha_{\mathrm{loc}}$ for all $0<\alpha<1$), except perhaps at a discrete set of singular points. We do this by showing that much of the free boundary solves an obstacle problem whose endogenous obstacle is $C^2$. From a slightly stronger conclusion, we deduce the free boundary becomes locally $C^\infty$ outside a closed set whose relative interior is empty. In response to the circulation of the present manuscript, we received a concurrent but independent work of Chen, Figalli and Zhang who verify a strengthening sufficient for this partial regularity result; (they show in particular that $\alpha=1$ and the discrete set mentioned above is empty).