Bi-Sobolev mappings and elliptic equations in the plane

Suppose that f=(u,v) is a homeomorphism in the plane of the Sobolev class such that its inverse is of the same Sobolev class. We prove that u and v have the same set of critical points. As an application we show that u and v are distributional solutions to the same non-trivial degenerate elliptic equation in divergence form. We study similar properties also in higher dimensions.