Solutions of the differential inequality with a null lagrangian: higher integrability and removability of singularities. I

The aim of this paper is to derive the self-improving property of integrability for derivatives of solutions of the differential inequality with a Lagrangian. More precisely, we prove that the solution of the Sobolev class with some Sobolev exponent slightly smaller than the natural one determined by the structural assumption on the involved Lagrangian actually belongs to the Sobolev class with some Sobolev exponent slightly larger than this natural exponent. We also apply this property to improve H\"older regularity and stability theorems of [19].