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

AbstractWe prove that a Pfaff system with coefficients in Llocp, p>2, in a simply-connected open subset Ω of R2 has at least a nontrivial solution of class Wloc1,p(Ω) provided that its coefficients satisfies a compatibility condition in the distributional sense. If in addition the set Ω 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 Wloc1,p and Llocp, with p>2, and satisfy together the Gauss and Codazzi–Mainardi equations in the distributional sense