Green's formulas with little regularity on a surface - Application to Donati-like compatibility conditions on a surface

In this Note, we establish two Green's formulas with little regularity on a surface. These formulas are then used for identifying and justifying Donati-like compatibility conditions on a surface, guaranteeing that the components of two symmetric matrix fields (c_αβ) and (r_αβ) with c_αβ and r_αβ in the space L²(ω), where ω is a domain in R2, are the covariant components of the linearized change of metric and linearized change of curvature tensors associated with a displacement vector field of a surface θ(ω-), where θ:ω-→R3 is a smooth immersion. © 2013 Académie des sciences.