On the Geometric Rigidity interpolation estimate in thin bi-Lipschitz domains

Abstract

This work is concerned with developing asymptotically sharp geometric rigidity estimates in thin domains. A thin domainΩ in space is roughly speaking a shell with non-constant thickness around a regular enough two dimensional compact surface. We prove a sharp geometric rigidity interpolation inequality that permits one to bound the Lp distance of the gradient of a u ∈W 1,p field from any constant proper rotation R , in terms of the average Lp distance (nonlinear strain) of the gradient from the rotation group, and the average Lp distance of the field itself from the set of rigid motions corresponding to the rotation R . The constants in the estimate are sharp in terms of the domain thickness scaling. If the domain mid-surface has a constant sign Gaussian curvature then the inequality reduces the problem of estimating the gradient ∇u in terms of the nonlinear strain ∫ Ωdist p (∇u(x),SO(3))dx to the easier problem of estimating only the vector field u in terms of the nonlinear strain with no asymptotic loss in the constants. This being said, the new interpolation inequality reduces the problem of proving “any” geometric one well rigidity problem in thin domains to estimating the vector field itself instead of the gradient, thus reducing the complexity of the problem. Funding. This material is based upon work partially supported by the National Science Foundation under Grants No. DMS-1814361, and partially supported by the Regents’ Junior Faculty Fellowship 2018 by UCSB. Manuscript received 8th February 2019, revised 6th June 2020 and 19th June 2020, accepted 18th June 2020.