Non-Continuity of Perron Solution

• Non-Continuity of the Perron Solution is characterized by the breakdown of classical harmonic solutions on domains with cusp singularities, despite smooth boundary data.
• Variational and Sobolev-space methods underpin explicit energy minimization approaches that yield Perron solutions in irregular domains like Lebesgue's.
• The study reveals non-local boundary effects where even isolated changes in data can trigger discontinuities at singular cusp points.

Lebesgue's Domain refers to a specific class of domains in potential theory, most prominently exemplified by a three-dimensional region bounded by surfaces of revolution around a thin line segment (a "rod") with variable mass density. The notion arises from the analysis of the Dirichlet problem for partial differential equations on non-smooth domains, particularly those with inward-pointing cusp singularities, and plays a pivotal role in understanding non-local and generic failures of boundary regularity in harmonic theory.

1. Geometric and Analytic Definition

Let $S = \{(0, 0, z) : 0 \leq z \leq 1\}$ denote the interval on the $z$-axis in $\mathbb{R}^3$. The domain $D = \mathbb{R}^3 \setminus S$ is considered, and the Newtonian potential of a "thin rod" $S$ with mass density $\rho(z)$ is defined at each $(r, z)$ (with $r = \sqrt{x^2 + y^2}$) by

$$V(r, z) = \int_{0}^{1} \frac{\zeta}{\sqrt{r^2 + (\zeta - z)^2}}\,d\zeta,$$

where, in Lebesgue's original example, $\rho(\zeta) = \zeta$. For any fixed $c > 0$, the level set $\{V(r, z) = c\}$ defines an analytic curve $L_c$ in the $(r, z)$-halfplane, which may be parametrized as $\{(r_c(z), z)\}$ for $z$ in some interval $(z_1, z_2)$ with $r_c(z_1) = r_c(z_2) = 0$ and $r_c(z) \geq 0$.

Lebesgue's domain $\Omega \subset \mathbb{R}^3$ is formed as the region between two such surfaces of revolution for levels $0 < c_1 < 1 < c_2$: $$\Omega = \{ (r, \theta, z) : c_1 < V(r, z) < c_2 \}.$$ This domain exhibits an inward-pointing cusp at the origin $(0,0,0)$ and is bounded by two connected components, $\Gamma_{c_1} = \{V = c_1\}$ and $$\Gamma_{c_2} \cup \{(0,0,0)\} = \{V = c_2\} \cup \{\text{cusp tip}\}$$ (Arendt et al., 17 Dec 2025).

2. Potential Theory and Boundary Structure

The potential $V$ is smooth and harmonic throughout $D = \mathbb{R}^3 \setminus S$. The boundary of $\Omega$ in the $xz$-plane ($y = 0$) consists of two curves $x = \pm r_{c_1}(z)$ and $x = \pm r_{c_2}(z)$ with the inner curves meeting at the cusp point $(0, 0)$. For a general rod density $\rho$ with $\rho(0) = 0$, the potential remains smooth off $S$. In the case of $\rho(\zeta) = \zeta$, explicit integration yields

$$V(r,z) = z\ln\bigl(\sqrt{(1-z)^2 + r^2} + 1 - z \bigr) - z\ln\bigl(\sqrt{z^2 + r^2} - z\bigr) + \sqrt{(1-z)^2 + r^2} - \sqrt{z^2 + r^2}.$$

The monotonicity and analytic structure of the level sets are ensured by Lemma 4.1 in (Arendt et al., 17 Dec 2025), which confirms that $r \mapsto V(r, z)$ is strictly decreasing for each fixed $z$.

3. Variational and Classical Dirichlet Solutions

Let $\varphi \in C(\partial\Omega)$ be continuous on the boundary of a bounded domain $\Omega \subset \mathbb{R}^3$. The variational formulation considers the Sobolev space $H^1(\Omega)$ and seeks $u_\varphi$ as the minimizer of the Dirichlet energy

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

subject to the boundary condition $u_\varphi|_{\partial\Omega} = \varphi$ in the sense of traces. An explicit variational solution is constructed as $u_\varphi = \Phi - v$, where $\Phi$ continuously extends $\varphi$ and $\Delta v = \Delta\Phi$ in the sense of distributions, with $v \in H^1_0(\Omega)$ [(Arendt et al., 17 Dec 2025), Theorem 2.1]. This solution coincides with the Perron solution, which is built as the infimum of all supersolutions dominating $\varphi$ [(Arendt et al., 17 Dec 2025), Theorem 3.2].

Crucially, on Lebesgue's domain, classical harmonic solutions—those continuous up to the entire boundary—may not exist even for boundary data $\varphi \in C^\infty(\partial\Omega)$. This non-existence results from the domain's failure to be Dirichlet-regular at the inward-pointing cusp, as formalized in Proposition 4.2 and Corollary 4.3 of (Arendt et al., 17 Dec 2025).

4. Singular Boundary Behavior and Non-Local Regularity

A defining feature of Lebesgue's domain is the generic and non-local nature of discontinuity in harmonic extensions at the cusp. Theorem 5.1 of (Arendt et al., 17 Dec 2025) asserts that if $z_0$ is a singular boundary point (such as the cusp), then any non-trivial change in the boundary data—even at a point far from $z_0$—can destroy continuity of $u_\varphi$ at $z_0$. In fact, the set of boundary data for which the solution remains continuous at a singularity is meagre in the sense of Baire category (Corollary 5.2). This demonstrates the robustness and non-locality of singular boundary behavior in domains like Lebesgue's.

A plausible implication is that attempts to guarantee classical (i.e., continuous) solutions for all boundary data must take domain regularity into account, as even smooth data cannot compensate for geometric singularities at the boundary.

5. Main Results and Characterizations

Key results relevant to Lebesgue's domain include:

• Analyticity of Boundary Curves: For each fixed $c > 0$, the boundary curves $L_c$ arising from level sets of $V$ are single analytic graphs parametrized in $z$ (Arendt et al., 17 Dec 2025).
• Dirichlet Energy Minimization: If $u_\varphi \in H^1(\Omega)$, then

$$u_\varphi = \arg\min\{E[w]: w \in H^1(\Omega), w|_{\partial\Omega} = \varphi\}$$

whenever the minimum is finite [(Arendt et al., 17 Dec 2025), Theorem 2.7].

• Variational Solution Equals Classical When Possible: If $\varphi$ admits a classical harmonic solution $u$, then $u = u_\varphi$; otherwise, the variational/Perron solution need not be continuous throughout $\bar{\Omega}$ [(Arendt et al., 17 Dec 2025), Proposition 2.3].
• Vasilesco’s Characterization: A bounded harmonic function $u$ on $\Omega$ is the Perron solution of $\varphi$ if and only if $u(x) \to \varphi(z)$ as $x \to z$ quasi-everywhere on $\partial\Omega$ [(Arendt et al., 17 Dec 2025), Theorem 3.2].
• Non-decibility of Classical Solvability: There exist simple (even piecewise-constant) boundary data $\varphi$ for which the Dirichlet problem admits no classical solution due to the cusp singularity [(Arendt et al., 17 Dec 2025), Proposition 4.2, Corollary 4.3].

6. Significance in Potential Theory and Analysis

Lebesgue's domain serves as a canonical example in classical potential theory illustrating the breakdown of local criteria for boundary regularity. It highlights the distinction between variational and classical solutions in domains with geometric singularities and reveals the generically non-local effect that singular points have on boundary behavior.

The study of Lebesgue's domain also establishes the utility of Sobolev-space and variational methods in handling boundary value problems where classical approaches fail due to irregularity. A key conceptual outcome is that the existence of classical solutions cannot be assured solely by boundary data regularity when the domain exhibits cusp-like singularities. The findings illuminate core aspects of non-locality and Baire-generic properties in the regularity theory of harmonic and elliptic PDEs (Arendt et al., 17 Dec 2025).