On Pfaff systems with LP coefficients and their applications in differential geometry

We prove that a Pfaff system with coefficients in L-loc(p), p > 2, in a simply-connected open subset to ohm of R-2 has at least a nontrivial solution of class W-loc(1,p)(ohm) provided that its coefficients satisfies a compatibility condition in the distributional sense. If in addition the set ohm is connected, the Cauchy problem associated with the Pfaff system has a unique solution. An application of this result is that the fundamental theorem of surface theory holds under the assumption that the first and second fundamental forms are respectively of class W-loc(1,p) and L-loc(p), with p > 2, and satisfy together the Gauss and Codazzi-Mainardi equations in the distributional sense. (c) 2005 Elsevier SAS. All rights reserved.