TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 355, Number 5, Pages 1987–2009 S 0002-9947(03)03237-9 Article electronically published on January 8, 2003 METRIC CHARACTER OF HAMILTON–JACOBI EQUATIONS ANTONIO SICONOLFI Abstract. We deal with the metrics related to Hamilton–Jacobi equations of eikonal type. If no convexity conditions are assumed on the Hamiltonian, these metrics are expressed by an inf–sup formula involving certain level sets of the Hamiltonian. In the case where these level sets are star–shaped with respect to 0, we study the induced length metric and show that it coincides with the Finsler metric related to a suitable convexification of the equation. Introduction This paper is devoted to the (nonsymmetric) distances of R N related to the Hamilton–Jacobi equations of eikonal type H (x,Du)=0 (I) with continuous Hamiltonian H satisfying the inequality H (x, 0) < 0 for any x. For H convex in the second variable, the recognition of the metric character of equation (I) is a central point in Hamilton–Jacobi theory, see [10]. This issue has been revised by Kruzkov [13] and P. L. Lions [15] and other authors using weak notions of solutions. In the first part of the paper we study the convex case, which for us means (see [8], [9]) that we just assume the convexity of the level sets Z (x)= {p : H (x,p) ≤ 0}. This is of course different from requiring the convexity of H as previously done. We also weaken slightly the regularity assumption on the Hamiltonian (see section 2). In this case the related distance L is then the value function of the variational problem inf  1 0 δ(ξ, ˙ ξ )dt, (II) where δ(x,q) is the support function of Z (x) at q and the infimum is taken with respect to all Lipschitz continuous curves defined in [0, 1] with fixed endpoints. In section 2 we prove that for any fixed y 0 , L(y 0 , ·) is a viscosity solution of (I) in R N  {y 0 }. Although this fact is generally known, at least when H is convex, our proof is based on a different point of view. In fact it comes directly from the analysis of the local behaviour of L and from its convexity property. Received by the editors May 9, 2000 and, in revised form, May 18, 2001. 2000 Mathematics Subject Classification. Primary 35F20, 49L25. Key words and phrases. Hamilton–Jacobi equations, viscosity solutions, distance functions. Research partially supported by the TMR Network “Viscosity Solutions and Applications”. c 2003 American Mathematical Society 1987 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use