arXiv subjects
Publications and source records attributed to Lucas D. O'Brien.
In this paper, we study the Rochet-Choné model of monopolist pricing in the seller's market limit. We introduce a monotonicity formula which yields an asymptotic description of solutions, and prove that the proportion of consumers priced out of the market vanishes as overall consumer demand increases, providing a counterpoint to Armstrong's desirability of exclusion.
For a compactly supported probability measure $μ$ on the $d$-dimensional space $\mathbb{R}^d$, the average distance problem asks us to minimize the average distance functional over all compact, connected, $Σ\subseteq \mathbb{R}^d$ satisfying the Hausdorff $1$-measure constraint $\mathcal{H}^1(Σ) \leq \ell$. This problem was first introduced in 2002 by Buttazzo, Oudet, and Stepanov to study optimal transport problems with free regions on which the transport cost vanishes, and has undergone a considerable amount of research since. Most recently, Kobayashi, Kim, and the author studied the structure of these regions using the barycentre field, a tool for studying the average distance functional introduced previously by Kobayashi, Hayase, and Kim. In this paper, we build upon this work to prove in much greater generality a topological description of minimizers of the average distance problem conjectured by Buttazzo, Oudet, and Stepanov. In particular, we prove this conjecture in all dimensions in the case originally studied by these authors.
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 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 $Ω$ 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 $C^2$. 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 $α=1$ and the discrete set mentioned above is empty).
A proof of the following theorem is given, answering an open problem attributed to Kunen: suppose that $T$ is compact and that $Y$ is the image of $X$ under a perfect map, $X$ is normal, and $Y\times T$ is normal. Then $X \times T$ is normal.