Torsion transplantation and a uniform reverse Kohler-Jobin inequality for Robin problems

Henry Zweiman. September 15, 2026.

An eight-page research preprint proposing an affirmative answer to the planar reverse Robin Kohler-Jobin question of Buttazzo, Cito and Solombrino. Balls uniquely maximize the first Robin eigenvalue times a sufficiently large power of torsional rigidity among all bounded planar Lipschitz sets of fixed area. The exponent is uniform in the positive Robin parameter and area. The same proof gives a nonlinear extension for 1<p<=n/(n-1), linear spectral-deficit control and equivalent product/torsion deficits.

Prepared with OpenAI Codex. The originating assistant completed an internal hand-proof and primary-source audit. No independent human review, certified priority or journal acceptance is claimed. The assessment supports proceeding as one substantial preprint under the revised research standard; it is a revisable judgment.

The Talenti comparisons of Alvino--Nitsch--Trombetti and Amato--Gentile--Masiello and the Faber--Krahn theorem of Bucur--Daners are essential credited inputs. The new step is the Robin torsion-composition trial and the resulting parameter-uniform competing-product estimate. The linear Robin case in dimensions at least three, the optimal threshold and the forward Robin question remain unresolved here. The known linear Dirichlet reverse inequality is not counted as another contribution.

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