Existence of C^1 critical subsolution of the Hamilton-Jacobi equation. inventiones mathematicae.

We consider an Hamiltonian 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 ?(x, p) ? T?M, H(x, p) ? Kp ?C? (K) ; (2) (Uniform boundedness in the fibers) for every R ? 0, we have A? (R) = sup{H(x, p) | p ? R} < +? ; (3) (Strict convexity in the fibers) for every (x, p) ? T?M, the second derivative along the fibers ?2H/?p2(x, p) is positive definite. We obtain Theorem 1.1. Under assumptions (1) to (3) above, if there is a global subsolution u : M ? R of H(x, dxu) = c, then there is a global C1 subsolution v : M ?R. In fact, it is possible to show that there exists c[0] ? R, such that H(x, dxu) = c admits no subsolution for c < c[0] and has subsolutions for c ? c[0]. The constant c[0] will be called the critical value, or the Mañ´e critical value. We will say that u : M ? R is a critical subsolution if it is a subsolution of H(x, dxu) = c[0]. It can even be shown that the equation H(x, dxu) = c[0] admits a viscosity solution (see below for definition), see for example [12] or [7]. Moreover it is also well-known that for c > c[0], there exists C? global subsolutions u : M ?R of H(x, dxu) = c, see [6], for the compact case, or the appendix of [12] for the most general case. So in fact the new result is the following. Theorem 1.2. There exists a C1 subsolution u : M ? R of H(x, dxu) = c[0]