Discontinuous solutions in \(L^\infty\) for Hamilton-Jacobi equations
This paper is concerned with the global discontinuous solutions in \(L^\infty\) of the Cauchy problem for Hamilton-Jacobi equations \[ \begin{cases} u_t+ H(t,x,u,Du)=0,\;x\in\mathbb{R}^n,\;0\leq t\leq T,\\ u(0,x)= \varphi(x). \end{cases} \] The approach introduced to construct global discontinuous solutions allows the initial data only in \(L^\infty\) and applies to the equations with nonconvex Hamiltonians. The profit functions are introduced to formulate the notion of discontinuous solutions in \(L^\infty\). The existence of global discontinuous solutions in \(L^\infty\) is established by taking the infimum of the supsolution set or the supremum of the subsolution set. These solutions in \(L^\infty\) coincide with the viscosity solutions and the minimax solutions, provided that the initial data are continuous. A prototypical equation is studied to analyze the stability of discontinuous solutions in \(L^\infty\). The analysis also shows that the global discontinuous solutions are determined by the topology in which the initial data are approximated.
scientific article; zbMATH DE number 1462259