Continuity in Fréchet topologies of a surface as a function of its fundamental forms

A generalization due to Sorin Mardare of the fundamental theorem of surface theory for surfaces with little regularity asserts that, if, for any p>2, the components of a positive-definite 2×2 symmetric matrix field in the space Wloc1,p and the components of another 2×2 symmetric matrix field in the space Llocp satisfy together the Gauss and Codazzi-Mainardi equations in a simply-connected open subset of R2, then there exists a surface defined in the three-dimensional Euclidean space E3 by an immersion with components in the space Wloc2,p, whose first and second fundamental forms are precisely the given matrix fields; besides, this surface is uniquely determined up to isometries in E3.