We investigate the value function of the Bolza problem of the Calculus of Variations
$$V(t,x)=inf\left\{{\int }_0^{t}L(y\left(s\right),y^{\text{'}}\left(s\right))ds+\varphi \left(y\left(t\right)\right):y\in W^{1,1}(0,t;{ℝ}^n),y\left(0\right)=x\right\},$$ with a lower semicontinuous Lagrangian and a final cost $\varphi$, and show that it is locally Lipschitz for whenever is locally bounded. It also satisfies Hamilton-Jacobi inequalities in a generalized sense. When the Lagrangian is continuous, then the value function is the unique lower semicontinuous solution to the corresponding Hamilton-Jacobi equation, while for discontinuous Lagrangian we characterize...

We consider minimization problems of the form ${\min }_{u\in \varphi +W_0^{1,1}\left(\Omega \right)}{\int }_{\Omega }[f\left(Du\left(x\right)\right)-u\left(x\right)]\mathrm{d}x$ where $\Omega \subseteq {ℝ}^N$ is a bounded convex open set, and the Borel function $f:{ℝ}^N\to [0,+\infty ]$ is assumed to be neither convex nor coercive. Under suitable assumptions involving the geometry of and the zero level set of , we prove that the viscosity solution of a related Hamilton–Jacobi equation provides a minimizer for the integral functional.