On generalized Jacobians in the sense of Clarke for the inverse of a bi‐Lipschitz map and applications in relaxation theory

Abstract Relaxed variational problems in the scalar case involving functions are frequently characterized by the convex envelope of the energy density . For functions with superlinear growth and Lipschitz continuous gradient, a formula for is presented in an open neighborhood of any maximal phase simplex provided that is and strictly convex close to the vertices of the phase simplex. This formula involves a carefully chosen parameterization of which identifies those subsets in in which may be written as a convex combination of a prescribed number of function values.