Quantitative symmetry in a mixed Serrin-type problem for a constrained torsional rigidity

We consider a mixed boundary value problem in a domain contained in a half-ball and having a portion of its boundary in common with the curved part T of the half-ball. The problem has to do with some sort of constrained torsional rigidity. In this situation, the relevant solution u satisfies a Steklov condition on T and a homogeneous Dirichlet condition on the rest S of the boundary . We provide an integral identity that relates (a symmetric function of) the second derivatives of the solution u in the domain to its normal derivative on S. A first significant consequence of this identity is a rigidity result under a quite weak overdetermining integral condition for the normal derivative on S: in fact, it turns out that S must be a spherical cap that meets T orthogonally. This result returns the one obtained by Guo and Xia under the stronger pointwise condition that the values of the normal derivative be constant on S. A second important consequence is a set of stability bounds, which quantitatively measure how S is far uniformly from being a spherical cap, if the normal derivative deviates from a constant in the Lebesgue 1-norm.