A characteristic initial boundary value problem for a symmetric positive system
Let \(\Omega\) be a bounded open subset of \(\mathbb{R}^n\) with smooth boundary \(\partial\Omega\). In \(\Omega\) the authors consider a first-order symmetric system \[ L[u]= \sum^n_{j= 1} A_j(x) \partial_j u+ B(x) u, \] \(A_j(x)\), \(B(x)\in C^\infty(\overline\Omega)\), \(A^*_j(x)= A_j(x)\) with \(u= (u_1,\dots, u_N)\) and \(\partial_j= {\partial\over \partial x_j}\). For \(x\in \partial\Omega\), \(A_b(x)= \sum^n_{j= 1} \nu_j A_j(x)\) in the boundary matrix where \(\nu= (\nu_1,\dots, \nu_n)\) is the unit outward normal to \(\Omega\). The case is considered that \(A_b(x)\) changes the definiteness simply crossing an embedded \(n- 2\)-dimensional submanifold of \(\partial\Omega\). The authors discuss the existence of regular solutions to the boundary value problem.
scientific article; zbMATH DE number 872595