Existence of C 1 critical subsolutions of the Hamilton-Jacobi equation

Abstract

Let M be a C∞ second countable manifold without boundary. We denote by TM the tangent bundle and by π : TM → M the canonical projection. A point in TM will be denoted by (x, v)with x ∈ M and v ∈ Tx M = π−1(x). In the same way a point of the cotangent space T ∗M will be denoted by (x, p) with x ∈ M and p ∈ T ∗ x M, a linear form on the vector space Tx M. We will suppose that g is a complete Riemannian metric on M. For v ∈ Tx M, the norm ‖v‖ is g(v, v)1/2. We will denote by ‖ · ‖ the dual norm on T ∗ x M. We will also use the notations ‖v‖x , for v ∈ Tx M, and ‖p‖x , for v ∈ T ∗ x M. We will assume in the whole paper that H : T ∗M → R is a function of class at least C2, which satisfies the following three conditions (1) (Uniform superlinearity) for every K ≥ 0, there exists C∗(K ) ∈ R such that