Nonlinear Korn Inequalities on a Hypersurface

The authors establish several estimates showing that the distance in W^1,p, 1 < p < ∞, between two immersions from a domain of R ⁿ into Rⁿ⁺¹ is bounded by the distance in L ^p between two matrix fields defined in terms of the first two fundamental forms associated with each immersion. These estimates generalize previous estimates obtained by the authors and P. G. Ciarlet and weaken the assumptions on the fundamental forms at the expense of replacing them by two different matrix fields.