Resolve the Robin overdetermined-domain conjecture for the torsion problem

Determine whether every smooth bounded domain that is a critical point, under a volume constraint, of the Robin torsion energy for the Laplacian must be a ball and whether the corresponding solution must be radial, without restricting the overdetermined constant to the special value treated in the partial result.

Background

The paper studies a Serrin-type overdetermined problem for the p-Laplacian with Robin boundary conditions. In the Laplacian case p=2, the relevant overdetermined boundary condition arises from the shape derivative of the Robin torsion energy under a volume constraint. The cited conjecture concerns the rigidity of critical domains for this variational problem: namely, whether such a domain must be a ball with a radially symmetric torsion solution.

The authors explain that a previous result established this rigidity only under the additional restriction that the overdetermined constant equals the special value C~0\widetilde C_0. They describe that result as a first step toward resolving the conjecture, so the unrestricted rigidity assertion remains unresolved in the cited discussion.

References

Analogously to the Serrin's result , in the authors took a first step toward resolving that conjecture by proving that, if $\Tilde{C} = \Tilde{C}_0$, where $\Tilde{C}_0 = -R{2} - \frac{R}{\beta} (N + 1)$, then $\Omega$ is a ball and $u$ is radial.

— Serrin-type overdetermined problem for the $p$-Laplacian with Robin boundary conditions  (2609.29393 - Barbato et al., 24 Sep 2026) in Section 1, Introduction, subsection “Overview”