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

This work is concerned with developing asymptotically sharp geometric rigidity estimates in thin domains. A thin domain $\Omega $ 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 $L^p$ distance of the gradient of a $\mathbf{u}\in W^{1,p}$ field from any constant proper rotation $\mathbf{R}$, in terms of the average $L^p$ distance (nonlinear strain) of the gradient from the rotation group, and the average $L^p$ distance of the field itself from the set of rigid motions corresponding to the rotation $\mathbf{R}$. 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 $\nabla \mathbf{u}$ in terms of the nonlinear strain $\int _\Omega \mathrm{dist}^p(\nabla \mathbf{u}(x),SO(3))\mathrm{d}x$ to the easier problem of estimating only the vector field $\mathbf{u}$ 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.