2022/10/19 by Rolando Magnanini, Magnanini, Rolando, Giorgio Poggesi +1 · 2 citations
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Composite Material Mechanics #Elasticity and Wave Propagation #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2210.10288
openalex publication_date 2022/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a mixed boundary value problem in a domain Ω contained in a half-ball B+ and having a portion T of its boundary in common with the curved part of ∂ 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 Σ= ∂Ω∖ T ⊂ B+. We provide an integral identity that relates (a symmetric function of) the second derivatives of the solution in Ω to its normal derivative uν on Σ. A first significant consequence of this identity is a rigidity result under a quite weak overdetermining integral condition for uν on Σ: in fact, it turns out that Σ 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 uν be constant on Σ. A second important consequence is a set of stability bounds, which quantitatively measure how Σ is far uniformly from being a spherical cap, if uν deviates from a constant in the norm L1(Σ).