Overdetermined systems of nonlinear partial differential equations (Q1113355)

Former results concerning compatible systems of two-dimensional partial differential equations (cf. the preceding review) are generalized to the higher-dimensional case \((\partial /\partial t)Z(t)=A(Z)\), \(Z(0)=z\), \(Z=(Z_ 1,...,Z_ n)\), \(z=(z_ 1,...,z_ n)\), \(t=(t_ 1,...,t_ m)\), \(\partial /\partial t=(\partial /\partial t_ 1,...,\partial /\partial t_ m)^ T\) with the zero vector 0 and an \(m\times n\)-matrix A. In case of rank \((A)=m\) there exist two mutually inverse vector functions F(z),f(z): \(C^ n\to C^ n\), so that the solution has locally the structure \(Z=F(tC_{mn}+f(z))\) with \(C_{mn}=(I_ m,O_{m,n-m})\). In case of rank \((A)=r<m\) with \(A=\left( \begin{matrix} A_ r\\ BA_ r\end{matrix} \right)\), where \(A_ r\) is the submatrix of the first r rows of A, and \(t=(u,v)\) with \(u=(t_ 1,...,t_ r)\), \(v=(t_{r+1},...,t_ m)\), the solution has the structure \[ Z=F((u+vB(z))C_{rn}+f(z))\quad with\quad B(Z)=B(z). \]