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

Abstract

We consider a mixed boundary value problem in a domain $\Omega$ contained in a half-ball $B_{+}$ and having a portion $\bar{T}$ of its boundary in common with the curved part of $\partial B_{+}$. 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 $\Sigma =\partial \Omega \setminus \bar{T}\subset B_{+}$. We provide an integral identity that relates (a symmetric function of) the second derivatives of the solution in $\Omega$ to its normal derivative $u_{\nu }$ on $\Sigma$. A first significant consequence of this identity is a rigidity result under a quite weak overdetermining integral condition for $u_{\nu }$ on $\Sigma$: in fact, it turns out that $\Sigma$ must be a spherical cap that meets $T$ orthogonally. This result returns the one obtained by J. Guo and C. Xia under the stronger pointwise condition that the values of $u_{\nu }$ be constant on $\Sigma$. A second important consequence is a set of stability bounds, which quantitatively measure how $\Sigma$ is far uniformly from being a spherical cap, if $u_{\nu }$ deviates from a constant in the norm $L^1(\Sigma )$.

The article has been accepted for publication in Calculus of Variations and Partial Differential Equations. This amended version includes various improvements and incorporates the referee's suggestions

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