A new arXiv preprint reports a sharply defined mathematical result: every weakly stable solution of Serrin’s problem, under the conditions stated in the paper, is a ball of radius 1. The result applies in every Euclidean dimension n≥2, and the proof also rules out multiple components, making the full domain connected.
The finding is a classification theorem, not an experiment. The objects under study are domains—regions in Euclidean space—and smooth functions defined on them. The authors prove what shape such a domain must have when the function satisfies a particular equation, boundary behavior and weak-stability inequality.
The shape hidden in the equations
Serrin’s problem ties the behavior of a function inside a domain to strict conditions on its edge. In a domain Ω, the function u must satisfy −∆u=n, remain positive, and equal zero on ∂Ω. Its gradient must also have magnitude 1 on the boundary. The paper asks whether a domain supporting such a function, together with weak stability, must be compact and round rather than taking some other shape.
Weak stability is defined through an inequality applied to compactly supported smooth test functions whose boundary integral is zero. Full stability would apply that inequality to all such test functions. The distinction is central: the theorem reaches its conclusion using the weaker condition.
The proof follows the geometry
The argument first establishes strict bounds for the function and its gradient. Throughout Ω, the estimates give |Du|<1 and 0<u<1. In ordinary terms, the function stays between zero and one inside the domain, while its slope remains strictly below the boundary gradient prescribed by Serrin’s problem.
The proof also controls how the boundary bends, establishing the mean-curvature bound H≤n on ∂Ω. That estimate gives the authors geometric information about the boundary as they examine whether the domain can remain large or split into separate pieces.
The analysis then separates finite-volume from infinite-volume possibilities. In the infinite-volume case, the authors derive large-radius area–volume asymptotics and use a balancing argument to obtain the stability information required for a contradiction. The broader proof is deterministic, drawing on gradient estimates, the Bochner formula, maximum principles, the Harnack inequality, divergence theorems, cutoff functions and stability inequalities.
A key step concerns what happens along a selected sequence of radii tending to infinity. The volume added by a radial shell becomes negligible relative to the relevant boundary area, while the ratio nV/A tends to 1. The assertion is made along that sequence; it is not a uniform statement about every sufficiently large radius.
From one component to the whole domain
For a connected solution with finite volume, the paper’s classification is direct: the domain is a ball of radius 1. This result supplies the central geometric conclusion once the finite-volume and connectedness conditions are in place.
The authors then rule out a domain with two or more components. After the components are classified individually as balls, weak stability produces a contradiction if more than one is present. The domain must therefore be connected, so the componentwise conclusions yield the classification of the full solution.
Together, these steps give the main theorem: for every dimension n≥2, a weakly stable solution satisfying the stated Serrin conditions is a ball of radius 1. The result also extends the stated earlier two-dimensional theorem to all dimensions and removes the assumption that the boundary has bounded curvature.
What the theorem does—and does not—cover
The conclusion is conditional on a specific mathematical setup. It applies to smooth domains and smooth solutions of the stated overdetermined boundary-value problem, with the paper’s weak-stability requirement. It does not classify arbitrary solutions when that stability assumption is absent.
The equation itself is also part of the scope. The result is tied to the stated −∆u=n problem and should not be read as a classification for arbitrary nonlinearities or unrelated boundary problems. The supplied account likewise leaves open how far comparable conclusions might extend under weaker regularity or different stability assumptions.
The document is an arXiv version 1 manuscript dated 20 Aug 2026. Because the work is a deterministic proof about mathematical domains and functions, it provides no empirical effect estimate or statistical validation.
The authors are Michael Eichmair and Thomas Koerber. The research was funded in whole or in part by the Austrian Science Fund, through grants 10.55776/PAT1307525 and 10.55776/PAT2423724. The authors state that AI tools suggested neither the arguments nor the text; Claude Opus 5 was used for typesetting checks.
Paper data and sources
Original title: Weakly stable solutions of Serrin's problem
Authors: Michael Eichmair, Thomas Koerber
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text