Abstract
We consider a boundary value problem for an elliptic differential operator of order 2m in a domain \({\mathcal{D}} \subset {\mathbb{R}}^{n}\). The boundary of \({\mathcal{D}}\) is smooth outside a smooth manifold Y of dimension 0 ≤ q < n − 1, and \(\partial {\mathcal{D}}\) bears edge type singularities along Y . The Lopatinskii condition is assumed to be fulfilled on the smooth part of \(\partial {\mathcal{D}}\). The corresponding spaces are weighted Sobolev spaces \(H^{{s,\gamma }} {\left( {\mathcal{D}} \right)}\), and this allows one to define ellipticity of weight γ for the problem. The resolvent of the problem is assumed to possess rays of minimal growth. The main result says that if there are rays of minimal growth with angles between neighbouring rays not exceeding π(γ + 2m)/n, then the root functions of the problem are complete in \(L^{2} ({\mathcal{D}})\). In the case of second order elliptic equations the results remain true for all domains with Lipschitz boundary.
Published Version
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