An Integro-Extremization Approach for Non Coercive and Evolution Hamilton-Jacobi Equations

We devote the integro-extremization method to the study of the Dirichlet problem for homogeneous Hamilton-Jacobi equations $$\begin{cases} F(Du)=0 & \quad \textrm{in} \quad\O\cr u(x)=\varphi(x) & \quad \textrm{for} \quad x\in \partial \O, \end{cases}$$ with a particular interest for non coercive hamiltonians $F$, and to the Cauchy-Dirichlet problem for the corresponding homogeneous time-dependent equations $$\begin{cases} \frac{\partial u}{\partial t}+ F(\nabla u)=0 & \quad \textrm{in} \quad ]0,T[\times \O\cr u(0,x)=\eta(x) & \quad \textrm{for} \quad x\in\O \cr u(t,x)=\psi(x) & \quad \textrm{for} \quad (t,x)\in[0,T]\times \partial \O. \end{cases}$$ We prove existence and some qualitative results for viscosity and almost everywhere solutions, under suitably convexity conditions on the hamiltonian $F$, on the domain $\O$ and on the boundary datum, without any growth assumptions on $F$