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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.

##### 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