New estimates of the distance between two surfaces in terms of the distance between their fundamental forms

A nonlinear Korn inequality on a surface is any estimate of the distance, up to a proper isometry of R³, between two surfaces measured by some appropriate norms (the “left-hand side” of the inequality) in terms of the distances between their three fundamental forms measured by some appropriate norms (the “right-hand side” of the inequality). The first objective of this paper is to provide several extensions of a nonlinear Korn inequality on a surface obtained in 2006 by the first and third authors and Gratie, then measured by means of H¹-norms on the left-hand side and L¹-norms on the right-hand side. First, we extend this inequality to W^1,p-norms on the left-hand side and L^q-norms on the right-hand side for any p > 1 and q ≥ 1 that satisfy p/2 ≤ q ≤ p; second, we show how the third fundamental forms can be disposed in the right-hand side; and third, we show that there is no need to introduce proper isometries of R³ in the left-hand side if the surfaces satisfy appropriate boundary conditions. The second objective is to provide nonlinear Korn inequalities on a surface where the left-hand sides are now measured by means of W^2,p-norms while the right-hand sides are measured by means of W^1,p-norms, for any p > 3. These nonlinear Korn inequalities on a surface themselves rely on various nonlinear Korn inequalities in a domain in Rⁿ, recently obtained by the first and third authors in 2015 and by the first author and Sorin Mardare in 2016.