A surface in W2,p is a locally Lipschitz-continuous function of its fundamental forms in W1,p and Lp, p > 2

The fundamental theorem of surface theory asserts that a surface in the three-dimensional Euclidean space E-3 can be reconstructed from the knowledge of its two fundamental forms under the assumptions that their components are smooth enough classically in the space C-2(omega) for the first one and in the space C-1(omega) for the second one and satisfy the Gauss and Codazzi-Mainardi equations over a simply-connected open subset w of R-2; the surface is then uniquely determined up to proper isometries of E-3. Then S. Mardare showed in 2005 that this result still holds under the much weaker assumptions that the components of the first form are only in the space W-loc(2,p) (omega) and those of the second form only in the space L-loc(p)(omega), the components of the immersion defining the reconstructed surface being then in the space W-loc(2,p) (omega), p > 2. The purpose of this paper is to complement this last result as follows. First, under the additional assumption that w is bounded and has a Lipschitz -continuous boundary, we show that a similar existence and uniqueness theorem holds with the spaces W-m,W-P(omega) instead of W-loc(m,p)(omega). Second, we establish a nonlinear Korn inequality on a surface asserting that the distance in the W-2,W-p (omega)-norm, p > 2, between two given surfaces is bounded, at least locally, by the distance in the W-1,W-P(omega)-norm between their first fundamental forms and the distance in the L-P(omega)-norm between their second fundamental forms. Third, we show that the mapping that uniquely defines in this fashion a surface up to proper isometries of E-3 in terms of its two fundamental forms is locally Lipschitz-continuous. (C) 2018 Elsevier Masson SAS. All rights reserved.