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

Abstract

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 Wloc1,p(ω) and those of the second form only in the space Llocp(ω), the components of the immersion defining the reconstructed surface being then in the space Wloc2,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 Wm,p(ω) instead of Wlocm,p(ω). Second, we establish a nonlinear Korn inequality on a surface asserting that the distance in the W2,p(ω)-norm, p>2, between two given surfaces is bounded, at least locally, by the distance in the W1,p(ω)-norm between their first fundamental forms and the distance in the Lp(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.