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The nonhomogeneous boundary problem for quasilinear positive symmetric systems. (Chinese. English summary) Zbl 0533.35013

Summary: In this paper we discuss the nonhomogeneous boundary problems for the following quasilinear positive symmetric systems: \[ \sum^{n}_{i=1} A^ i(x,u)\partial u/\partial x^ i+au = (1/b)f(x,u)\quad\text{in }\Omega, \tag{1} \]
\[ -\beta_- u|_{\Gamma} = (1/K)g(x,u). \tag{2} \] We make the following assumptions: (i) \({\bar \Omega}\) is a closed bounded domain in \(R^ n\), \(\Gamma\) is a sufficiently smooth boundary which allows corner points. (ii) \(\Gamma\) is noncharacteristic, the boundary condition (2) is stable admissible and all coefficients are smooth enough. (iii) The conditions of consistency are satisfied up to the order \(2m+\rho -1\) \[ (m=[n/2]+1),\quad \partial^ i f(x,0)|_{\beta_-} = 0, \quad \partial^ i g(x,0)|_{\beta_-} = 0 \quad(i\leq 2m+\rho -1). \] (iv) \(a\) is large enough, after \(a\) is determined, \(b\) and \(K\) are sufficiently large.

MSC:

35F30 Boundary value problems for nonlinear first-order PDEs
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