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Hamilton-Jacobi equations having only action functions as solutions

Authors:

Email author" target="_blank">Thomas?Str?mberg Email author

Institution:

(1) Department of Mathematics, Luleå University of Technology, SE-971 87 Luleå, Sweden

Abstract:

Let L(x, v) be a Lagrangian which is convex and superlinear in the velocity variable v, and let H(x, p) be the associated Hamiltonian. Conditions are obtained under which every viscosity solution $u \in C((0,T] \times \mathbb{R}^n )$ of the Hamilton-Jacobi equation

$u_t (t,x) + H(x,\,\nabla u(t,x)) = 0\quad {\text{in }}(0,T] \times \mathbb{R}^n$

is an action function in the large, i.e.,

$u(t,x) = {\text{inf}}\{ u(0,\,X(0)) + \int\limits_0^t {L(X(\tau ),\,\dot X(\tau ))d\tau {\text{: }}X \in W^{1,1} (0,t;\mathbb{R}^n ),\,X(t) = x\} }$

for all $(t,x) \in (0,T] \times \mathbb{R}^n .$ Received: 13 June 2003

Keywords:

Mathematics Subject Classification (1991)" target="_blank">Mathematics Subject Classification (1991) 35F25 49L25

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