We consider the problem of the body of minimal resistance as formulated in [2], Sect. 5: minimize F (u) :where Ω is the unit disc of R 2 , in the class of radial functions u : Ω → [0, M] satisfying a geometrical property (1), corresponding to a single-impact assumption (M > 0 is a given parameter). We prove the existence of a critical value M * of M . For M ≥ M * , there exist a unique local minimizer of the functional. For M < M * , the set of local minimizers is not compact in H 1 , though they all achieve the same value of the functional.

The present article is devoted to the study of a constrained weighted total variation minimization problem, which may be viewed as a relaxation of a generalized Cheeger problem and is motivated by landslide modeling. Using the fact that the set of minimizers is invariant by a wide class of monotone transformations, we prove that level sets of minimizers are generalized Cheeger sets and obtain qualitative properties of the minimizers: they are all bounded and all achieve their essential supremum on a set of positive measure.

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Abstract. This article deals with the numerical computation of the Cheeger constant and the approximation of the maximal Cheeger set of a given subset of R d . This problem is motivated by landslide modelling as well as by the continuous maximal flow problem. Using the fact that the maximal Cheeger set can be approximated by solving a rather simple projection problem, we propose a numerical strategy to compute maximal Cheeger sets and Cheeger constants.Mathematics Subject Classification. 49Q10, 65K10.

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