A note on the area and coarea formulas for general volume densities and some applications (Q5965202)

The paper is comprehensive investigation of extended classical problems. The generalizations of the classical change of variables formula and Fubini theorem were proved by Federer. The authors take another look at these formulas and rewrite them for general volume densities. Their results involve a certain formalism for which one has not found any precise reference in the literature. One can relate a volume density on a Finsler manifold to the Riemannian density, via the Radon-Nykodim derivative, but sometimes it is convenient to be able to employ the analogues of the well-known Riemannian formulas directly to the intrinsic densities that might arise. To emphasize their potential use, the authors consider several analytic and geometric applications such as the tube formula for hypersurfaces, the Sobolev inequality and the first variation of area, all of them in the anisotropic world. For example, the general coarea formula presented in this paper allows one essentially to repeat the classical proof of \textit{H. Federer} and \textit{W. H. Fleming} [Ann. Math. (2) 72, 458--520 (1960; Zbl 0187.31301)] of the Euclidean Sobolev inequality to the anisotropic world. The paper consists of six sections. In Section 2 the authors present the linear picture. In Section 3 they state the analogues of the formula of a change of variables and the Fubini theorem. In Section 4 they present the area and coarea formulas for Lipschitz maps between Euclidean spaces. The proofs of the area and coarea theorems for general densities utilize their well-known Riemannian analogues and some canonical relations between the correction factors derived in Section 2. In Section 5 the authors derive the tube formula and introduce a natural anisotropic connection on an oriented hypersurface. They show that the first variation of the anisotropic area is the integral of the divergence with respect to this connection and includes a version of the divergence theorem relevant to this framework. Section 6 is dedicated to the proof of the anisotropic Sobolev inequality together with the aforemendoned result about anisotropic Minkowski content.