On Pfaff systems with [formula omitted] coefficients and their applications in differential geometry

Abstract

We prove that a Pfaff system with coefficients in L loc p , p > 2 , in a simply-connected open subset Ω of R 2 has at least a nontrivial solution of class W loc 1 , 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 W loc 1 , p and L loc p , with p > 2 , and satisfy together the Gauss and Codazzi–Mainardi equations in the distributional sense.