Serrin's overdetermined theorem and weak Bernoulli laws without Alt--Caffarelli regularity

Technical Summary of "Serrin's Overdetermined Theorem and Weak Bernoulli Laws Without Alt–Caffarelli Regularity"

Problem Statement
The paper investigates the rigidity of Serrin-type overdetermined boundary value problems and one-phase free boundary problems when the classical regularity assumptions of Alt–Caffarelli theory are removed. Specifically, the author studies domains $\Omega \subset \mathbb{R}^n$ supporting a solution $u$ to the distributional Bernoulli law:
$$\Delta u = c \mathcal{H}^{n-1}\lfloor \partial^*\Omega - f(u)\mathbf{1}_\Omega dx \quad \text{in } \mathcal{D}'(\mathbb{R}^n),$$
where $f \in C^2$, $f(0) > 0$, and $\partial^*\Omega$ denotes the reduced boundary. The central question is whether the distributional identity alone, without the uniform growth or density hypotheses (specifically the uniform upper density bound $\mathcal{H}^{n-1}(\partial\Omega \cap B_r(x)) \leq L r^{n-1}$) required in the Alt–Caffarelli theory, forces $\Omega$ to be a ball (in the case $f \equiv 1$) or a disk (in the planar case).

Methodology
The paper employs a combination of complex analysis, harmonic analysis, and weighted elliptic PDE techniques to construct counterexamples and prove rigidity results.

1. Planar Rigidity (Smirnov Class):
   For planar domains, the author establishes that if $\Omega$ is a Smirnov domain (a class of Jordan domains where the conformal map's derivative is an outer function), the distributional Bernoulli law implies $\Omega$ is a disk. The proof utilizes the harmonic quadrature identity derived from the Bernoulli law to construct a boundary Cauchy-orthogonality relation. This relation yields a bounded holomorphic function in the unit disk, allowing the application of Weinberger's $P$-function method in a low-regularity setting.

2. Planar Counterexamples (Non-Smirnov):
   To show the sharpness of the Smirnov condition, the author constructs non-spherical, non-Smirnov domains satisfying the law. The construction relies on the Duren–Shapiro–Shields (DSS) mechanism, which generates domains with non-trivial singular inner factors in their conformal derivatives.

   • Obstruction: A naive perturbation of the disk using a fixed singular measure fails to satisfy the strong quadrature identity (Proposition 4.1).
   • Resolution: The author introduces a four-fold symmetry to eliminate the Fourier modes associated with disk translations (which would otherwise require Lyapunov–Schmidt reduction). A nonlinear fixed-point argument is then used to solve for an outer factor that corrects the boundary density, while the singular inner factor remains to ensure the domain is not Smirnov.

3. Higher-Dimensional Counterexamples ($n \geq 3$):
   The planar conformal construction does not directly extend to higher dimensions. Instead, the author uses an axially symmetric reduction.

   • Weighted Reduction: The problem is reduced to a weighted planar conductivity problem with a weight $r^p$ (where $p=n-2$) arising from the Laplacian in cylindrical coordinates.
   • Singular Perturbation: The construction involves a weighted elliptic trace operator. A key difficulty is the "sin $t$" mode obstruction generated by the axial weight. The author introduces a second singular parameter $\delta$ to create a controlled top-bottom imbalance in the singular measure, which cancels the axial forcing term.
   • Fixed Point: A contraction mapping argument solves for the high-frequency modes, while the scalar sine mode is balanced by adjusting $\delta$. The resulting meridian domain is rotated to form the $n$-dimensional counterexample.

Key Contributions and Results

• Theorem 1.1 (Planar Rigidity): If $\Omega \subset \mathbb{C}$ is a bounded Jordan domain with rectifiable boundary that is a Smirnov domain and satisfies the harmonic moment identity (equivalent to the distributional Bernoulli law for $f \equiv 1$), then $\Omega$ is a disk. This strengthens previous results by replacing the uniform upper density assumption with the weaker analytic Smirnov condition.
• Theorem 1.2 (Planar Counterexample): For any $f \in C^2$ with $f(0) > 0$, there exist bounded, rectifiable, non-Smirnov Jordan domains that are not disks but satisfy the distributional Bernoulli law. These domains possess a non-trivial singular inner factor in their conformal derivative, leading to a failure of uniform upper density bounds on the boundary.
• Theorem 1.4 (Higher-Dimensional Counterexample): For $n \geq 3$, there exist bounded sets of finite perimeter $\Omega \subset \mathbb{R}^n$ (homeomorphic to $B^2 \times S^{n-2}$) that satisfy the distributional Bernoulli law but are not balls. These sets fail the uniform upper density condition (1.3), demonstrating that the Alt–Caffarelli density hypothesis is essential for rigidity in dimensions $n \geq 3$.
• Regularity Thresholds: The paper identifies the Smirnov condition as the exact threshold for rigidity in the plane and the uniform upper density bound as the threshold for $n \geq 3$.

Significance
The paper demonstrates that the uniform growth and density assumptions in the classical Alt–Caffarelli theory are not merely technical conveniences but are necessary to exclude singular boundary behaviors that allow for non-spherical solutions.

• Failure of Distributional Uniqueness: The distributional Bernoulli law alone is insufficient to guarantee radial symmetry or free boundary regularity, even among rectifiable domains.
• Role of Singular Factors: The counterexamples rely on singular inner factors in conformal maps (in 2D) or singular measures in the construction (in $n$D). These factors are invisible to the boundary arclength density but affect the interior Jacobian density, thereby disrupting the harmonic moment identity required for rigidity.
• Sharpness of Existing Theorems: The results confirm that the rigidity theorems relying on uniform density bounds (such as those in [23]) are sharp; removing these bounds allows for the existence of "rough" solutions that satisfy the weak formulation but lack the geometric regularity of a ball.

The work clarifies the precise boundary between flexibility and rigidity in free boundary problems, showing that without specific regularity constraints (Smirnov in 2D, uniform density in $n \geq 3$), the overdetermined problem admits a rich class of non-spherical solutions.