Lebesgue's Domain in PDEs and Convex Geometry

• Lebesgue's Domain is a prototypical construct in PDE theory and convex geometry, defined by an axisymmetric region with a singular inward cusp causing non-local discontinuities.
• The domain illustrates how variational solutions to the Dirichlet problem fail to be continuous at the cusp, regardless of the smoothness of the boundary data.
• It also connects to the universal covering problem, revealing optimal convex covering properties and sharp metric bounds in high-dimensional geometry.

Lebesgue's Domain refers both to a prototypical domain in the theory of partial differential equations (PDE), critical for understanding boundary regularity phenomena, and to the classical universal covering problem in convex geometry, originally posed by Lebesgue. In the context of the Dirichlet problem, Lebesgue’s domain is an axisymmetric, simply-connected region in $\mathbb{R}^3$ constructed using the potential generated by a thin rod with a vanishing mass density at one end. The boundary of this domain includes a singular inward-pointing cusp, providing a canonical example where harmonic extensions (solutions to the Dirichlet problem) generically fail to be continuous at the cusp, regardless of the regularity of prescribed boundary data. In parallel, Lebesgue's universal covering problem seeks the smallest convex planar set that covers all sets of unit diameter by congruence, underlying a major open problem in geometric measure theory.

1. Definition and Construction of Lebesgue’s Domain

Lebesgue’s domain $\Omega$ arises from the potential theory of the Newtonian kernel in $\mathbb{R}^3$, associated to a "thin rod" segment $S = \{(0,0,z) : 0 \leq z \leq 1\}$, with linear mass density $\rho(z)=z$. The Newtonian potential induced by this rod is given by

$$V(r,z) = \int_0^1 \frac{\zeta}{\sqrt{r^2 + (\zeta - z)^2}} \, d\zeta,$$

where $r = \sqrt{x^2 + y^2}$ in cylindrical coordinates. For each fixed $c > 0$, the level set equation $V(r,z) = c$ defines an analytic curve $L_c$ in the $(r,z)$-plane, which can be parametrized as $$L_c = \{(r_c(z), z) : z_1 \leq z \leq z_2, V(r_c(z), z) = c\}$$ with $r_c(z) \geq 0$, $r_c(z_1)=r_c(z_2)=0$ and $r_c(z)$ real-analytic on $(z_1, z_2)$. For two levels $0 < c_1 < 1 < c_2$,

$$\Omega = \{(r, \theta, z) \in \mathbb{R}^3 : c_1 < V(r,z) < c_2\}$$

is the region between the surfaces of revolution $\{V = c_1\}$ and $\{V = c_2\}$; its boundary consists of two connected components, $\Gamma_{c_1}$ and $\Gamma_{c_2} \cup \{(0,0,0)\}$, the latter containing an inward-pointing cusp at the origin. In planar cross-section ($y=0$), $\Omega$ is bounded by the curves $x = \pm r_{c_1}(z)$, $x = \pm r_{c_2}(z)$, with the inner pair meeting at the cusp $(0,0)$ (Arendt et al., 17 Dec 2025).

2. Variational Formulation and the Dirichlet Problem

For a bounded domain $\Omega \subset \mathbb{R}^3$ with boundary data $\varphi \in C(\partial\Omega)$, the variational solution $u_\varphi$ minimizes the Dirichlet energy

$E[v] = \int_\Omega |\nabla v|^2\, dx$

over all $w \in H^1(\Omega)$ with $w|_{\partial\Omega} = \varphi$ in the Sobolev trace sense. If one chooses a continuous extension $\Phi \in C(\overline\Omega)$ of $\varphi$ such that $\Delta\Phi \in H^{-1}(\Omega)$, then there is a unique $v \in H^1_0(\Omega)$ with $\Delta v = \Delta\Phi$ in $H^{-1}(\Omega)$, and the variational solution is $u_\varphi = \Phi - v$. The Perron solution $\overline u$ (the supremum of all subharmonic functions bounded above by $\varphi$) coincides with the variational solution for any continuous boundary data; both are harmonic in $\Omega$ and attain boundary values in the Sobolev sense (Arendt et al., 17 Dec 2025).

A key consequence is that if a classical (continuous up to $\partial\Omega$) harmonic solution exists, it coincides with $u_\varphi$. However, for domains like Lebesgue's, where the boundary includes a cusp singularity, regularity up to the boundary typically fails, even for $C^\infty$ boundary data.

3. Boundary Regularity, Non-Locality, and Generic Discontinuity

Lebesgue’s domain demonstrates the non-decidiability of classical solvability in the Dirichlet problem: for boundary data $\varphi$ that is constant on each boundary component, the variational (and Perron) solution will take different values along $\Gamma_{c_1}$ and $\Gamma_{c_2} \cup \{\text{cusp}\}$, enforcing discontinuity at the cusp if $A \neq B$ (Arendt et al., 17 Dec 2025). No regularity assumption on $\varphi$ suffices to guarantee continuity at the cusp.

This failure is robust and non-local. Theorem 5.1 establishes that if $\Omega$ has a singular boundary point $z_0$ (the cusp), then any non-trivial change in $\varphi$ away from $z_0$ destroys continuity of $u_\varphi$ at $z_0$. In other words, the set of data $\varphi$ for which $u_\varphi$ is continuous at $z_0$ is meagre in $C(\partial\Omega)$ (Corollary 5.2), showing that discontinuity is generic in the Baire category sense. This property illustrates the fundamentally non-local influence of the boundary geometry on harmonic extension, as even distant perturbations of the boundary data away from the singular point impact regularity at the cusp.

4. Key Analytical Properties of Lebesgue’s Domain

Several technical properties underlie the singular behavior in Lebesgue’s domain:

• The potential $V(r,z)$ is smooth and harmonic in $D = \mathbb{R}^3 \setminus S$, with explicit analytic expressions for the level curves.
• For a general density $\rho : [0, L] \to [0, \infty)$ with $\rho(0) = 0$, the potential has the same regularity except at the supporting segment $S$.
• Lemma 4.1 confirms for each $z$ that $r \mapsto V(r,z)$ is strictly decreasing from $+\infty$ to $0$, and each level set $\{V(r,z)=c\}$ is a single analytic graph.
• The variational method always produces a unique global energy minimizer in $H^1(\Omega)$, consistent with the classical, if and only if a classical solution exists (Proposition 2.3, Theorem 2.7).

The domain $\Omega$ exhibits Dirichlet irregularity at the inward cusp point $(0,0,0)$: classical solutions cannot exist even for constant boundary data (Proposition 4.2, Corollary 4.3), a consequence of the geometric impossibility of harmonically bridging between two boundary components meeting at a singular tip.

5. Broader Context: Lebesgue’s Universal Covering Problem

Independently, Lebesgue's domain refers in convex geometry to the set-theoretic problem of universal covering. A universal cover in $\mathbb{E}^n$ is a measurable set $U$ such that for every set $A \subset \mathbb{E}^n$ of diameter $1$, there exists a congruent copy $\Phi(A)$ contained in $U$. Lebesgue's universal covering problem (1914) seeks the convex universal cover in the plane ($n=2$) of smallest area: $$\min\{\left(U\right) : U \subset \mathbb{R}^2 \ \text{convex, universal} \}.$$ Despite a century of research, the best lower bound is $0.832\dots$ and the best explicit construction achieves area $\approx 0.8440935944$ (Arman et al., 3 Dec 2025).

For general dimension, Jung's theorem provides the optimal exponential scale: any set of diameter $1$ in $\mathbb{E}^n$ is contained in a ball $J_n$ of radius $r_n = \sqrt{\frac{n}{2n+2}}$, forming a universal cover of volume $(J_n) = r_n^n (B_n)$. The result

$$(U) \geq \exp\left(-\sqrt{(\tfrac{5}{4} + o(1))n \ln n}\right) (J_n) = (1-o(1))^n (J_n)$$

shows that no universal cover is asymptotically smaller (in the exponential scale) than Jung's ball (Arman et al., 3 Dec 2025).

6. Consequences and Applications

Lebesgue’s domain in the PDE context exemplifies domains where the Sobolev-based variational approach extends the notion of solution beyond the classical, accommodating singular boundaries and non‐local effects. The robust non-locality and Baire-generic failure of boundary continuity at singular points such as cusps indicate the necessity of variational and weak solution concepts in many physical and geometric applications.

In convex geometry, the intractability of the minimal universal covering problem and the optimality of Jung’s solution in high dimension underscore the deep connections between metric properties, symmetries, and measure concentration phenomena in high-dimensional Euclidean spaces.

7. Key Results and Theoretical Insights

Property / Phenomenon	PDE/Dirichlet Domain ($\Omega$)	Universal Covering Problem
Singular Boundary	Cusp at $(0,0,0)$; not Dirichlet-regular	N/A
Regularity of Solution	Variational/Perron coincides, but fails continuity at cusp generically	Always exists (by construction); measure-minimizing cover is open
Role of Non-Locality	Data elsewhere affects regularity at singular point (Theorem 5.1)	Covering property is global
Main Optimal Structures	Energy minimizer in $H^1$ with given trace	Jung’s ball $J_n$ is optimal in exponential scale

Lebesgue's domain, in both analytic and geometric contexts, offers canonical settings where generic phenomena—such as the non-local destruction of boundary regularity or the irreducible size of universal covers—manifest in their sharpest form. These domains continue to serve as critical benchmarks for foundational advances in harmonic analysis, potential theory, and convex geometry (Arendt et al., 17 Dec 2025, Arman et al., 3 Dec 2025).