Existence of \(C^1\) critical subsolutions of the Hamilton-Jacobi equation (Q1879016)

\(M\) is a \(C^\infty\) second countable manifold without boundary, \(TM\) the tangent bundle and \(T^*M\) the cotangent space. Consider a function \(H: T^*M\rightarrow \mathbb R\) of class at least \(C^2\) which satisfies conditions of uniform superlinearity, of uniform boundedness in the fibers and of strict convexity in the fibers. Under these conditions, the authors prove two basic results: 1) if there is a global subsolution \(u: M \rightarrow \mathbb R\) of \(H(x,d_xu) = c\), then there is a global \(C^1\) subsolution of \(v: M \rightarrow \mathbb R\) ; 2) there exists a \(C^1\) global critical subsolution. The authors give other versions of result 2).