A Lower Bound for the Gradient of 1-harmonic Functions

We establish a lower bound for the gradient of the solution to 1-Laplace equation in a strongly star-shaped annulus with capacity type boundary conditions. The proof involves properties of the radial derivative of the solution, so that starshapedness of level sets easily follows. x1. Introduction In this paper we deal with solutions to the 1-Laplace equation 1 u = n X i;j=1 u x i u x j u x i x j = 0; (1 1) in a domain of R n. Equation (1 1) was rst considered by G. Aronsson ((Ar1], Ar2]) and naturally arises as the Euler equation of minimal Lipschitz extensions. It is a highly degenerate elliptic equation which is formally the limit, as p ! 1, of the p-Laplace equation p u = div(jDuj p?2 Du) = 0: (1 p) This limit process has been recently made rigorous by R. Jensen in J], where he establishes the fundamental result that any Dirichlet problem for equation (1 1) has a unique viscosity solution u 2 W 1;1 (() \ C 0 () which is the limit, as p ! 1, of the unique solution u p 2 W 1;p (() to equation (1 p) satisfying the same Dirichlet data (which exists and is unique by standard variational arguments), in the sense that u p ! u uniformly in and weakly in W 1;q (() for any q such that q < 1. For a discussion of the related concepts of absolutely minimizing Lipschitz extension, variational solution and viscosity solution to equation (1 1), we also refer to B-D-M]. Concerning the critical points of 1-harmonic functions, Aronsson proved that any non-constant C 2 solution to (1 1) in the plane has non-vanishing gradient ((Ar2]), this result has been recently extended to C 4 solutions in higher dimensions by L. C. Evans ((E]). On the other hand, Aronsson gave examples of C 1 non-constant (viscosity) solutions to (1 1) having an interior critical point ((Ar3]).