Solvability of systems of partial differential equations for functions defined on nonconvex sets (Q1592799)

The author considers some class of overdetermined systems of PDEs: \[ R(D_x,D_y)f=0,\;P(D_x)f=g \quad \text{in} \Omega\subset\mathbb R^{n+1},\tag{1} \] where \(R(D)\) is elliptic, \(f,g\in C^\infty(\Omega)\), \(g\) satisfies the compability condition \(R(D)g=0,\;x\in\mathbb R^n,y\in\mathbb R\), \(\Omega=\Omega_{N,\lambda}=\{(x,y)|\langle x,N\rangle>0\), \(|y|<\langle x,N\rangle^\lambda\}\), \(0\neq N\in\mathbb R^n,\;\lambda\geq 0\), \(m=\text{ord }P\). In contrast to the principle of Ehrenpreis-Palamodov which asserts that such a system can be solved on a convex open set, it is proved the following Theorem. If for any \(g\in C^\infty(\Omega_{N,\lambda})\) there is a solution \(f\in C^\infty(\Omega_{N,\lambda})\) of (1), then for any localization \(Q\) of \(P_m\) at \(\infty\) \[ Q(x+i\tau N)\neq 0,\;\forall (x,\tau)\in\mathbb R^n\times(\mathbb R\setminus \{0\}) \] if \(N\) is noncharacteristic for \(Q\). Especially, \(Q\) is then hyperbolic w.r.t. \(N\). A simple example: let \(\Delta_k\) be the Laplacian in \(k\) variables; then the system \(\Delta_4f=0\), \(\Delta_2f=g\) with \(\Delta_4g=0\) can be solved on \(\{x\in\mathbb R^4\mid x_1>0,|x_4|<x_1^\lambda\},\lambda\geq 0\), if and only if \(\lambda\leq 1\). For the proof the author uses his methods which were working in the problem of surjectivity for PDO on real analytic functions.