W^2,p -estimates for surfaces in terms of their two fundamental forms

Let p>2. We show how the fundamental theorem of surface theory for surfaces of class W^2,p_loc(ω) over a simply-connected open subset of R² established in 2005 by S. Mardare can be extended to surfaces of class W^2,p(ω) when ω is in addition bounded and has a Lipschitz-continuous boundary. Then we establish a nonlinear Korn inequality for surfaces of class W^2,p(ω). Finally, we show that the mapping that defines in this fashion a surface of class W^2,p(ω), unique up to proper isometries of E³, in terms of its two fundamental forms is locally Lipschitz-continuous.