Dirichlet problems for the 1-Laplace operator, including the eigenvalue problem, manuscript 2005

We consider a number of problems that are associated with the 1-Laplace operator Div (Du/|Du|), the formal limit of the p-Laplace operator for p → 1, by investigating the underlying variational problem. Since corresponding solutions typically belong to BV and not to W1,1, we have to study minimizers of functionals containing the total variation. In particular we look for constrained minimizers subject to a prescribed L1-norm which can be considered as an eigenvalue problem for the 1-Laplace operator. These variational problems are neither smooth nor convex. We discuss the meaning of Dirichlet boundary conditions and prove existence of minimizers. The lack of smoothness, both of the functional to be minimized and the side constraint, requires special care in the derivation of the associated Euler-Lagrange equation as necessary condition for minimizers. Here the degenerate expres-sion Du/|Du | has to be replaced with a suitable vector field z ∈ L ∞ to give meaning to the highly singular 1-Laplace operator. For minimizers of a large class of problems containing the eigenvalue problem, we obtain the surprising and remarkable fact that in general infinitely many Euler-Lagrange equations have to be satisfied. 1