Abstract

Under a rather large set of fairly complicated assumptions on the data, the authors extend the basic concepts and results from the theory of the so-called “minimax solutions” in [\textit{A. I. Subbotin}, Generalized solutions of first-order PDEs (Birkhäuser, Basel) (1994; Zbl 0820.35003)] to systems of nonlinear first order PDEs of the form: \[ {{\partial u_k}\over {\partial t}}(t,x)+H_k(t,x,u(t,x),\nabla_xu_k(t,x))=0 \forall (t,x)\in G:=(0,T)\times \mathbb{R}^n, k=1,2,\dots,m, \] \[ u(T,x):=(u_1(T,x),\dots, u_m(T,x))= u^0(x) \forall x\in \mathbb{R}^n. \] They introduce the concepts of minimax subsolutions, supersolutions and solutions for the system above and prove that a classical solution is a minimax solution and conversely, a minimax solution satisfies the system in the classical sense at each differentiability point. A theorem stating the existence of minimax solutions is also proved.