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

摘要

Let M be a C~∞r 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 ∈ T_xM = 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 T_xM. We will suppose the g is a complete Riemannian metric on M. For v ∈ T_xM, 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 ∈ T_x M, and ‖p‖_x for v ∈ T_x~* M.