A coarea-type formula for the relaxation of a generalized elastica functional

Abstract

We consider in R^2 the generalized elastica functional defined, for smooth functions, as the p-elastica energies of the level lines integrated over all levels. Extending the functional to L1, we study its L1-lower semicontinuous envelope and we prove that, for any u in BV, the envelope at u can be represented by a coarea-type formula involving suitable collections of W2,p curves that cover the essential boundaries of the level sets of u.