Fathi, Albert; Siconolfi, Antonio Existence of \(C^1\) critical subsolutions of the Hamilton-Jacobi equation. (English) Zbl 1061.58008 Invent. Math. 155, No. 2, 363-388 (2004). \(M\) 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). Reviewer: Ubiratan D’Ambrosio (São Paulo) Cited in 2 ReviewsCited in 81 Documents MSC: 37J50 Action-minimizing orbits and measures (MSC2010) 35F20 Nonlinear first-order PDEs 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games 70F20 Holonomic systems related to the dynamics of a system of particles 58D25 Equations in function spaces; evolution equations Keywords:Lagrangian systems; Hamilton-Jacobi equations PDFBibTeX XMLCite \textit{A. Fathi} and \textit{A. Siconolfi}, Invent. Math. 155, No. 2, 363--388 (2004; Zbl 1061.58008) Full Text: DOI