Lax–Oleinik Formula on Networks

We provide a Lax-Oleinik-type representation formula for solutions of time-dependent Hamilton-Jacobi equations, posed on network with rather general geometry, under standard assumptions the Hamiltonians. It depends given initial datum at $t=0$ and flux limiter vertices, which both have to be assigned in order problem uniquely solved. Previous results same direction are solely frame junction, namely single vertex. An important step get result is define suitable action functional prove existence as well Lipschitz-continuity minimizers between two fixed points time, despite fact that integrand lacks convexity vertices.