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On the solution of a boundary problem in a polygon. (Russian) Zbl 1137.35406
Let \(\Omega\) be a polygon \(\{x\in \mathbb R^2 : H^1\cdot x +d_1>0,\dots, H^n\cdot x +d_n>0\}\), bounded by the lines \(H^i\cdot x +d_i>0\), \(i=1,\dots,n\) with normal vectors \((H^i_1,H^i_2)\). The problem is to find \(u=(u_1(x_1,x_2),u_2(x_1,x_2))\), subject to the equation \[ Lu\equiv (a^1\cdot \nabla)\cdot\dots\cdot (a^n\cdot \nabla)u=0, \qquad x\in \Omega,\eqno(1) \] and the boundary conditions \[ u|_{\partial\Omega}=u'_{\nu}|_{\partial\Omega}=u^{(n-2)}_{\nu^{(n-2)}}|_{\partial\Omega}=0.\eqno(2) \] Here \(\nu\) is the outward normal to the boundary \(\partial\Omega\), \(\nabla=(\partial_{x_1},\partial_{x_2})\) and \(a^k=(a^k_1,a^k_2)\in C^2.\) The main result are necessary conditions for the existence of nontrivial solution of (1), (2).
35M99 Partial differential equations of mixed type and mixed-type systems of partial differential equations
14H70 Relationships between algebraic curves and integrable systems
35A05 General existence and uniqueness theorems (PDE) (MSC2000)