On the Hausdorff dimension and singularities of the monopolist's free boundary curve
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908920042225664 |
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| author | McCann, Robert J. O'Brien, Lucas D. Rankin, Cale |
| author_facet | McCann, Robert J. O'Brien, Lucas D. Rankin, Cale |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_14100 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Hausdorff dimension and singularities of the monopolist's free boundary curve McCann, Robert J. O'Brien, Lucas D. Rankin, Cale Analysis of PDEs 35R35 (Primary) 35Q91 90B50 91A65 91B41 91B43 (Secondary) 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). |
| title | On the Hausdorff dimension and singularities of the monopolist's free boundary curve |
| topic | Analysis of PDEs 35R35 (Primary) 35Q91 90B50 91A65 91B41 91B43 (Secondary) |
| url | https://arxiv.org/abs/2603.14100 |