Journal de Mathématiques Pures et Appliquées | 2019
A surface in W2,p is a locally Lipschitz-continuous function of its fundamental forms in W1,p and Lp, p > 2
Abstract
Abstract 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 ( ω ) for the first one and in the space C 1 ( ω ) for the second one—and satisfy the Gauss and Codazzi–Mainardi equations over a simply-connected open subset ω 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 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 E 3 in terms of its two fundamental forms is locally Lipschitz-continuous.