Hamilton–Jacobi equations with their Hamiltonians depending Lipschitz continuously on the unknown
Hamilton–Jacobi equations with their Hamiltonians depending Lipschitz continuously on the unknown
We study the Hamilton–Jacobi equations H(x,Du,u)=0 in M and ∂u/∂t+H(x,Dxu,u)=0 in M×(0,∞), where the Hamiltonian H=H(x,p,u) depends Lipschitz continuously on the variable u. In the framework of the semicontinuous viscosity solutions due to Barron–Jensen, we establish the comparison principle, existence theorem, and representation formula as value functions for extended real-valued, …