A surface in W^2,p is a locally Lipschitz-continuous function of its fundamental forms in W^1,p and L^p, p  > 2

The fundamental theorem of surface theory asserts that a surface in the three-dimensional Euclidean space E³ can be reconstructed from the knowledge of its two fundamental forms under the assumptions that their components are smooth enough—classically in the space C²(ω) for the first one and in the space C¹(ω) for the second one—and satisfy the Gauss and Codazzi–Mainardi equations over a simply-connected open subset ω of R²; the surface is then uniquely determined up to proper isometries of E³. 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^1,p (ω) and those of the second form only in the space L_loc^p (ω), the components of the immersion defining the reconstructed surface being then in the space W_loc^2,p (ω), p >2. The purpose of this paper is to complement this last result as follows. First, under the additional assumption that ω is bounded and has a Lipschitz-continuous boundary, we show that a similar existence and uniqueness theorem holds with the spaces W^m,p (ω) instead of W_loc^m,p (ω). Second, we establish a nonlinear Korn inequality on a surface asserting that the distance in the W^2,p (ω)-norm, p >2, between two given surfaces is bounded, at least locally, by the distance in the W^1,p (ω)-norm between their first fundamental forms and the distance in the L^p(ω)-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 E³ in terms of its two fundamental forms is locally Lipschitz-continuous.