Generalized solutions of partial differential equations of the first order. The invariance of graphs relative to differential inclusions
The paper is devoted to the determination of minimax solutions for first order partial differential equations of the form \[ F(x,u,Du)=0, \qquad x\in G\subset \mathbb{R}^n, \] by means of techniques from differential game theory. The paper has more a review character pointing out some approaches to the notion of generalized (minimax) solution. One defines the minimax, upper and lower solutions for the above equation. The minimax solution turns out to be in the same time an upper and lower solution. The connection between \(u\)-stable resp. \(v\)-stable functions and upper resp. lower solutions for the Hamilton-Jacobi-Bellman-Isaacs equation is shown. Also, the relation between minimax and viscosity solutions in the sense of Lions is largely discussed. The last two paragraphs contain results regarding discontinuous solutions for the Dirichlet boundary value problem: \[ H(x,Du)- u=0, \quad x\in G\subset \mathbb{R}^n, \qquad u(x)= \sigma(x), \quad x\in\partial G. \] One defines the upper, lower, and minimax solutions and one proves, by means of Lyapunov function, the uniqueness of a minimax solution.
first order partial differential equations
scientific article; zbMATH DE number 903434
scientific article