A mathematical preprint has identified a clean benchmark for a problem in partial differential equations: on bounded Lipschitz domains in n-dimensional space, the Bogovskii constant—a number defined by a variational problem for the divergence equation—is never below n, and balls attain that lower value. The result makes balls minimizers within the class studied, although it does not settle whether they are the only shapes that do so.
The authors present the work as a unified treatment of the divergence equation, the Bogovskii constant and related functional inequalities. Its proof framework uses functional decomposition to connect a weak variational formulation with a generalized Stokes system, bringing several analytical questions into one route.
What the lower bound means
At the core is a two-level variational problem. The inner infimum—the search for the best admissible vector field—is not merely approached: it is attained by one unique minimizing vector field. When the boundary has class C^{m+1,1} and the source belongs to H^m with zero mean, that privileged solution has H^{m+1} regularity.
That result feeds the paper’s main inequality. For every bounded Lipschitz domain, C_B is at least n, and the theorem supplies two sufficient criteria for strict inequality. Because balls reach the bound, n is not just a universal floor; it is a value that is actually achieved.
The analysis also asks when the outer variational supremum—the upper value sought in the associated optimization—is itself attained. If a domain has a C^4 boundary and C_B is greater than 2, an eigenfunction at the extremal Schur-complement value exists, and both stated variational suprema are attained. This theorem does not cover arbitrary Lipschitz domains.
Shape changes the answer
For balls, the conclusion is exact: C_B(B)=n. Since no bounded Lipschitz domain can have a smaller value, balls minimize the constant across that class. The paper leaves open whether the ball is the unique minimizing shape.
Ellipsoids show how geometry can move the result away from that benchmark. The paper gives an explicit privileged-solution analysis for a suitable n-dimensional source space, along with upper and lower bounds that depend on the ellipsoid’s axes. No proper ellipsoid—that is, no non-ball ellipsoid—attains C_B=n.
The lower bound also forces divergence in increasingly thin ellipsoids: when the maximum-to-minimum axis ratio tends to infinity, C_B tends to infinity. This is an asymptotic consequence of the stated bound, not a general exact formula for ellipsoids in every dimension.
Annuli follow the same broad pattern. For A_r^R with 0<r<R, the paper states that C_B(A_r^R)>n, and it gives a quantitative lower bound for the divergent thinning limit. Here too, the general result is a lower bound rather than a complete exact formula.
The pattern extends to higher order
The authors then extend the framework beyond the basic setting. The higher-order version has its own privileged solution, which attains the inner minimum, comes with a priori bounds and satisfies a higher-order generalized Stokes formulation. The corresponding constant is defined in the specified higher-order function spaces.
For the stated higher-order indices, the bound remains C_B^(m) at least n on bounded Lipschitz domains. The unit ball supplies the equality case as well: in dimensions n≥2, its higher-order Bogovskii constant equals n. Together, these results preserve the same dimension-based benchmark at higher order.
The boundaries of the result
The conclusions are tied to the assumptions used to obtain them. The setting requires bounded Lipschitz domains, while some attainment results require stronger boundary smoothness; the higher-order statements also depend on the chosen function spaces and trace conditions. The analysis therefore gives exact benchmarks and bounds, not an exact Bogovskii constant for every domain.
The work is an arXiv version-1 preprint dated 26 August 2026. Open questions include whether outer-supremum attainment extends to every Lipschitz domain, whether balls are the only minimizers, and whether sharper or exact higher-dimensional formulas can be obtained for ellipsoids.
Paper data and sources
Original title: A unified approach to the divergence equation and related functional inequalities
Authors: Filippo Gazzola, Hans-Christoph Grunau, Gianmarco Sperone
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: 10.13140/rg.2.2.24025.89448
Original paper · Full text