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

Abstract

A nonlinear Korn inequality on a surface is any estimate of the distance, up to a proper isometry of [Formula: see text], 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 [Formula: see text]-norms on the left-hand side and [Formula: see text]-norms on the right-hand side. First, we extend this inequality to [Formula: see text]-norms on the left-hand side and [Formula: see text]-norms on the right-hand side for any [Formula: see text] and [Formula: see text] that satisfy [Formula: see text]; 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 [Formula: see text] 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 [Formula: see text]-norms while the right-hand sides are measured by means of [Formula: see text]-norms, for any [Formula: see text]. These nonlinear Korn inequalities on a surface themselves rely on various nonlinear Korn inequalities in a domain in [Formula: see text], recently obtained by the first and third authors in 2015 and by the first author and Sorin Mardare in 2016.