The Asymptotically Sharp Geometric Rigidity Interpolation Estimate in Thin Bi-Lipschitz Domains
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The Asymptotically Sharp Geometric Rigidity Interpolation Estimate in Thin Bi-Lipschitz Domains D. Harutyunyan1
Received: 9 December 2019 © Springer Nature B.V. 2020
Abstract This work is part of a program of development of asymptotically sharp geometric rigidity estimates for thin domains. A thin domain in three dimensional Euclidean space is roughly a small neighborhood of regular enough two dimensional compact surface. We prove an asymptotically sharp geometric rigidity interpolation inequality for thin domains with little regularity. In contrast to that celebrated Friesecke et al. rigidity estimate [Commun. Pure Appl. Math. 55(11):1461–1506, 2002] for plates, our estimate holds for any proper rotations R ∈ SO(3). Namely, the estimate bounds the Lp distance of the gradient of any y ∈ W 1,p (, R3 ) field from any constant proper rotation R ∈ SO(3), in terms of the average Lp distance (nonlinear strain) of the gradient ∇y from the rotation group SO(3), and the average Lp distance of the field itself from the set of rigid motions corresponding to the rotation R. There are several remarkable facts about the estimate: 1. The constants in the estimate are sharp in terms of the domain thickness scaling for any thin domains with the required regularity. 2. In the special cases when the domain has positive or negative Gaussian curvature, the inequality reduces ´ the problem of estimating the gradient ∇y in terms of the prototypical nonlinear strain distp (∇y(x), SO(3))dx to the easier problem of estimating only the vector field y in terms of the nonlinear strain without any loss in the constant scalings as the Ansätze suggest. The later will be a geometric rigidity Korn-Poincaré type estimate. This passage is major progress in the thin domain rigidity problem. 3. For the borderline energy scaling (bending-to-stretching), the estimate implies improved strong compactness on the vector fields for free. Finally, this being said, our 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. Keywords Thin domains · Shells · Geometric rigidity · Korn’s inequality Mathematics Subject Classification 00A69 · 35J65 · 74B05 · 74B20 · 74K25
B D. Harutyunyan 1
Santa Barbara, USA
D. Harutyunyan
1 Introduction Let S ⊂ R3 be a connected and compact surface that has a normal a.e. (one may assume for instance that S is Lipschitz). Given a small parameter h > 0, recall that a shell S h of thickness h is the h/2 neighborhood of S in the normal direction, i.e., S h = {x + tn(x) : x ∈ S, t ∈ (−h/2, h/2)}, where for any point x ∈ S, the vector n(x) is the unit normal to S at x. Thin spatial domains are roughly shells with non-constant thickness. Namely, assume the functions g1h (x), g2h (x) : S → (0, ∞) are of order h Lipschitz functions, i.e., they fulfill the uniform conditions h ≤ g1h (x), g2h (x) ≤ c1 h,
and
|∇g1h (x)| + |∇g2h (x)| ≤ c2 h,
for a.e.
x ∈ S,
(1.1)
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