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On non-closure of range of values of elliptic operator for a plane angle. (English) Zbl 0835.35026

The following first-order system, elliptic in the sense of Douglis- Nirenberg is considered: \[ v+ \nabla u= F(x),\quad\text{div}(v)= 0,\quad\text{in } \Omega\subset \mathbb{R}^2.\tag{1} \] The main result of the paper states that the range of the operator \(L\) associated to (1) with either Dirichlet or Neumann boundary condition is dense but not closed in \(L^p(\Omega; \mathbb{R}^2)\), if \(\Omega= \{(r, \varphi)\mid r> 0, 0< \varphi< \alpha\}\) and \(p= 2/(1\pm \pi/\alpha)\).

MSC:

35F15 Boundary value problems for linear first-order PDEs
47F05 General theory of partial differential operators

Keywords:

angular domain
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