Hamilton-Jacobi equations having only action functions as solutions
The objective of this paper is to give an answer for the following problem: when can we conclude that the identity \[ u(t,x)= \text{inf}\Biggl\{u(0, X(0))+ \int^t_0 L(X(\tau),\dot X(\tau))\,d\tau: X\in W^{1,1}(0,t; \mathbb{R}^n), X(t)= x\Biggr\} \] holds for every viscosity solution \(u\in C((0, T]\times \mathbb{R}^n)\) of the Hamilton-Jacobi equation \[ u_t(t,x)+ H(x,\nabla u(t, x))= 0\quad\text{in }(0,T]\times\mathbb{R}^n,\tag{1} \] with no exception, and every \((t,x)\in (0,T]\times \mathbb{R}^n\)? The author proves that under conditions: (A) \(L\) is continuous in \(\mathbb{R}^n\times \mathbb{R}^n\); \(L(x,v)\) is a convex function of \(v\in\mathbb{R}^n\) for every \(x\in \mathbb{R}^n\) and \(L(x,v)\geq l(|v|)\) for all \((x,v)\in\mathbb{R}^n\times \mathbb{R}^n\) where \(l(s)/s\to\infty\) as \(s\to\infty\); (B) For some \({\mathcal T}> 0\) and every choice of \(x\in\mathbb{R}^n\) and \(\varepsilon\in (0,{\mathcal T})\), \(\lim_{|y|\to\infty} (S(t,x,y)- S(t+\varepsilon, x,y))=+\infty\) uniformly with respect to \(t\in (0,{\mathcal T}-\varepsilon)\), every viscosity solution \(u\) of (1) is an action function in the large.
scientific article; zbMATH DE number 2138531