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

Abstract: The simplest genuinely multidimensional monopolist's problem involves minimizing a linearly perturbed Dirichlet energy among nonnegative convex functions $u$ on an open domain $X \subset [0, \infty)<sup>2$. The geometry of the region of strict convexity $Ω\subset X$ for the unique minimizer $u$ is of central interest. A relatively closed portion $X_1⁰ \subset X$ 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 $X \subset \mathbf{R}<sup>2$, we build on results with Zhang to show that outside $X_1⁰ \cup {u=0}$, the free boundary of $Ω$ is a continuous curve of Hausdorff dimension one, and that $Ω$ has density $1/2$ along it (and is $C<sup>α_{\mathrm{loc}}$ 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 $C<sup>2$. For Rochet and Choné's square $X=(a,a+1)<sup>2$ with $a&gt;0$, 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 $C<sup>{2,\mathrm{Dini}}$, the free boundary becomes locally $C<sup>\infty$ outside a closed set whose relative interior is empty.

• The paper reduces the bunching–customization boundary to an obstacle problem by proving the constructed obstacle is C² whenever the ray-length function is continuous.
• The analysis shows singular free-boundary points are isolated because each has a unique quadratic blow-up and is a strict local maximum of the ray-length function.
• The results establish Hausdorff dimension at most one and bi-Hölder curve structure, with locally smooth regular components under Dini continuity and controlled accumulation for polygonal type spaces.

The paper studies the fine structure of the free boundary arising in the two-dimensional monopolist's problem of Rochet and Choné, as reformulated by McCann, Rankin, and Zhang (2603.14100). The monopolist's problem reduces to maximizing the profit functional

$$\Pi(u) = \int_X \left[ x \cdot \nabla u(x) - u(x) - \tfrac12 |\nabla u(x)|^2 \right] d\mathcal{H}^n$$

over the cone $\mathcal{U}$ of nonnegative convex functions $u \in W^{1,2}(\overline{X})$, where $X \subset [0,\infty)^2$ is the type space and $u$ is the indirect utility. The domain decomposes into an exclusion region $\Omega_0 = \{u = 0\}$ (buyers priced out), a customization region $\Omega_2$ on which $u$ is strictly convex and satisfies $\Delta u = 3$, and a bunching region $\Omega_1$ foliated by rays along which $\mathcal{U}$0 is affine. The boundary between $\mathcal{U}$1 and $\mathcal{U}$2 is the object of study: it is determined endogenously by the solution, making the problem a genuine free boundary problem.

Reduction to an obstacle problem

The analysis concentrates on tame free boundary points: points $\mathcal{U}$3 in $\mathcal{U}$4 whose leaf $\mathcal{U}$5 meets $\mathcal{U}$6 at a single smooth point $\mathcal{U}$7 where the outward normal distortion $\mathcal{U}$8 is strictly positive. A local foliation lemma shows that a neighbourhood of any tame ray is itself foliated by tame rays, yielding bi-Lipschitz $\mathcal{U}$9 coordinates adapted to the foliation.

In these coordinates, the authors construct $u \in W^{1,2}(\overline{X})$0, the minimal convex extension of $u \in W^{1,2}(\overline{X})$1 off the bunching region, and set $u \in W^{1,2}(\overline{X})$2. The key structural result is that $u \in W^{1,2}(\overline{X})$3 solves an obstacle problem

$u \in W^{1,2}(\overline{X})$4

with contact set $u \in W^{1,2}(\overline{X})$5 and free boundary $u \in W^{1,2}(\overline{X})$6. Crucially, whereas prior work only established that the obstacle was $u \in W^{1,2}(\overline{X})$7 — insufficient for Caffarelli's classical theory, which requires $u \in W^{1,2}(\overline{X})$8 obstacles — the paper proves a regularity upgrade: if $u \in W^{1,2}(\overline{X})$9 has modulus of continuity $X \subset [0,\infty)^2$0, then $X \subset [0,\infty)^2$1 has modulus bounded by $X \subset [0,\infty)^2$2. Since $X \subset [0,\infty)^2$3 is continuous, Heine–Cantor uniform continuity yields $X \subset [0,\infty)^2$4. This "two degrees of regularity gain" over $X \subset [0,\infty)^2$5 is what licenses the use of Blank's theory for obstacle problems with merely continuous right-hand side.

Discreteness of the singular set

Blank's alternative partitions the free boundary into regular points, where the density of the noncontact set tends to $X \subset [0,\infty)^2$6, and singular points, where the density of the contact set tends to zero. The paper's central contribution is showing that for the monopolist's problem the singular set is not merely small but discrete.

Two ingredients drive the argument. First, exploiting the ray structure, every blow-up at a singular point is unique and has the explicit form

$X \subset [0,\infty)^2$7

where $X \subset [0,\infty)^2$8 is normal to the ray through $X \subset [0,\infty)^2$9. Second, each singular point is shown to be a strict local maximum of the ray-length function $u$0: otherwise, a sequence of nearby rays of at least equal length would force the blow-up to vanish along a direction transverse to the ray, contradicting the explicit form above. Combining this with Blank–Hao's universal modulus $u$1 controlling the contact-set density at singular points, and a measure-stability lemma for blow-ups (symmetric-difference bound of order $u$2), a contradiction argument rules out accumulation of singular points at any tame free boundary point.

This improves the earlier result of McCann–Rankin–Zhang, which established only $u$3 without controlling the size of the singular set; discreteness echoes Monneau's analogous result for smoother obstacles.

Hölder curves and sharp dimension bounds

At regular points, Blank's Reifenberg-vanishing theorem states that the flatness modulus $u$4 tends to zero. Combined with the classical Reifenberg topological disk theorem, each connected component of the regular set admits an $u$5-bi-Hölder parametrization by an interval, for every $u$6, and hence has Hausdorff dimension at most one. Since the singular set is discrete, the full tame free boundary satisfies

$u$7

which is sharp. Under the additional hypothesis that $u$8 is Dini continuous — equivalent to Dini continuity of $u$9 — Blank's $\Omega_0 = \{u = 0\}$0 theorem applies, and a bootstrapping argument using the Lipschitz ray-direction field $\Omega_0 = \{u = 0\}$1 upgrades components of $\Omega_0 = \{u = 0\}$2 to locally $\Omega_0 = \{u = 0\}$3 curves outside a closed set with empty relative interior. The bootstrapping fails precisely where the free boundary tangent aligns with the ray direction; a Picard–Lindelöf argument shows such alignment cannot persist on an open set.

Global structure of the customization boundary

Under hypotheses (a)–(d), satisfied by all convex polygons $\Omega_0 = \{u = 0\}$4, the corollary extends these conclusions from the tame set $\Omega_0 = \{u = 0\}$5 to the entire free boundary $\Omega_0 = \{u = 0\}$6: its Hausdorff dimension is at most one, the closure of the singular set intersects $\Omega_0 = \{u = 0\}$7 countably, and singularities can accumulate only on the fixed boundary or on $\Omega_0 = \{u = 0\}$8. For Rochet and Choné's square $\Omega_0 = \{u = 0\}$9, the set $\Omega_2$0 consists of at most a single point $\Omega_2$1 on the diagonal, so singularities can accumulate only at $\Omega_2$2, at the ends of the analytic arc $\Omega_2$3, and at two limit points on $\Omega_2$4. The proof hinges on showing $\Omega_2$5 is connected — via reflection symmetry, concavity, and a concave-versus-convex graph argument limiting intersections to two — and that stray rays can produce at most two additional free-boundary components, whose interiors are analytic.

Limitations and open questions

Several hypotheses carry real weight. Assumption (c) requires $\Omega_2$6; while known for convex polygons, it is unverified for general convex domains, and the unpublished Caffarelli–Lions result gives only $\Omega_2$7 globally. Hypothesis (e), bounding $\Omega_2$8, is conjectured but verified only for the square, where the intersection is a single point. The Dini partial regularity statement is conditional: without Dini control on $\Omega_2$9, the Reifenberg-vanishing rate is unquantified, and Blank's example shows the free boundary may fail even to be locally a graph when $u$0 is merely continuous. The authors also note that the bi-Hölder parametrization does not imply Hölder or Dini continuity of $u$1 itself. Whether the singular set is finite rather than countable, and whether hypothesis (e) holds beyond the square, remain open.

Conclusion

The paper establishes that the monopolist's free boundary in the plane is, outside a relatively closed set of affine segments and the exclusion region, a curve of Hausdorff dimension one with density $u$2 along its regular part, with discrete singularities, and becomes smooth under a Dini condition. Methodologically, it demonstrates how an endogenous obstacle of only $u$3 regularity — gained from continuity of the ray length — suffices to import the low-regularity obstacle-problem machinery of Blank and Blank–Hao into contract theory, yielding quantitative geometric control of the customization region's boundary for economically relevant domains such as Rochet and Choné's square.