zbMATH — the first resource for mathematics

A symmetric positive system with nonuniformly characteristic boundary. (English) Zbl 1131.35324
The boundary value problem \(Mu =g\) on \(\partial \Omega \) for the linear system of \(k\) equations \(Lu = f\) on a bounded Lipschitz (or smooth) domain \(\Omega \subset \mathbb R^n\) is considered, where \(L = \sum ^n_{j=1} A_j\partial /\partial x_j + B\) with \(A_j, B\), and \(M\) being \((k\times k)\)-matrix valued functions of \(x\). In case of noncharacteristic boundary conditions, \(L^2\)- or \(H^1\)-solutions are proved to exist. For \(M(\cdot )\) not of a constant rank and alternating definiteness between positive and negative, existence of regular solutions is investigated, too.

35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35F15 Boundary value problems for linear first-order PDEs
35L50 Initial-boundary value problems for first-order hyperbolic systems