Generalized Cheeger sets related to landslides

We study the maximization problem, among all subsets X of a given domain Ω, of the quotient of the integral in X of a given function f by the integral on the boundary ofX of another function g. This is a generalization of the well-known Cheeger problem corresponding to constant functions f, g. The non-constant case is motivated by applications to landslides modeling where the the supremum given by a variational blocking problem appears as a safety coefficient. We prove that this coefficient is equal to the supremum of the shape optimization problem formerly mentioned. For constant data, this amounts to studying the first eigenvalue of the 1-laplacian operator. We prove existence of optimal sets, and give some differential characterization of their internal boundary. We study their symmetry properties using the Steiner symmetrization. In dimension two, we give explicit solutions for data depending only on one variable.