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Existence of \(C^1\) critical subsolutions of the Hamilton-Jacobi equation. (English) Zbl 1061.58008

\(M\) 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).

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
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