A symmetric positive system with nonuniformly characteristic boundary. (English) Zbl 1131.35324

Secchi, Paolo

Differ. Integral Equ. 11, No. 4, 605-621 (1998).

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.

Reviewer: Tomáš Roubíček (Praha)

MSC:

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