On the Hausdorff dimension and singularities of the monopolist's free boundary curve
Robert J. McCann, Lucas D. O'Brien, Cale Rankin
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 $Ω\subset X$ for the unique minimizer $u$ is of central interest. A relatively closed portion $X_1^0 \subset X$ of the domain is comprised of segments starting and ending on $\partial X$ along which $u$ is affine. For convex polygons and certain other 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 $Ω$ is a continuous curve of Hausdorff dimension one, and that $Ω$ has density $1/2$ along it (and is $C^α_{\mathrm{loc}}$ for all $0<α<1$), 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 $C^2$. For Rochet and Choné's square $X=(a,a+1)^2$ with $a>0$, there is a point $x_a$ on the diagonal such that $X_1^0\cap \partial Ω\subset \{x_a\}$, and the discrete singularities mentioned above can only accumulate at $x_a$ or at the two ends of the analytic arc $\{u=0\} \cap \partial Ω$, plus the two limit points of $X \cap \partial Ω$ on $\partial X$. Where the regularity of the endogenous obstacle can be improved to $C^{2,\mathrm{Dini}}$, the free boundary becomes locally $C^\infty$ outside a closed set whose relative interior is empty.