Bounded from below viscosity solutions of Hamilton-Jacobi equations (Q1386263)

The existence of a unique viscosity solution \(u\) of the equation \(u(x)+H(Du(x))=f(x)\), \(x\in\mathbb{R}^n\) such that \(u^-\) grows at most linearly, when \(f^-\) behaves analogously, \(H\) being nonlinear and convex, is proved. The results are extended to more general equations and applied to Dirichlet problems for unbounded domains. The results are sharpened, if the behaviour of \(H\) at infinity is prescribed and, in this case, the assumptions on \(f(x)\) are nearly optimal.