Right accessibility of semicontinuous initial data for Hamilton-Jacobi equations
The authors study the Cauchy problem for Hamilton-Jacobi equations \[ u_{t}+H(x,Du)=0, \text{ in } \mathbb{R}^{N}\times(0,T), u(x,0)=g(x), \text{ for } x\in\mathbb{R}^{N}, \] where the initial data \(g\) is an upper semicontinuous map from \(\mathbb{R}^{N}\) into the real line (that is, not necessarily continuous). Moreover, they impose the following hypothesis on the Hamiltonian \[ | H(x,p)-H(x,q)| \leq L| p-q| , \text{ for all } x,p,q\in \mathbb{R}^{N}, \] and \[ | H(x,p)-H(y,p)| \leq C_{R}(1+| p| )| x-y| , \text{ for all } | x| ,| y| \leq R,p\in \mathbb{R}^{N}. \] First, the authors establish that the generalized solution \(u(x,t)\) can be represented as the value function \(U(x,t)\) of a differential game. The formula is too much involved to be described here. Second, using this representation formula the authors obtain necessary and sufficient conditions in order for the solution to attain the initial data g through the so-called right accessibility, that is, along sequences \(\{t_{n}\}\), \(\{x_{n}\}\), with \(t_{n} >0\), \(x_{n}\in \mathbb{R}^{N}\), such that \(t_{n}\rightarrow 0\), \(x_{n}\rightarrow x_{0}\) and \(u(x_{n},t_{n})\rightarrow g(x_{0})\). It is worthly pointing out that no convexity assumptions on \(H\) are imposed.
scientific article; zbMATH DE number 2052054