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Approximation numbers of Sobolev imbeddings over unbounded domains
Authors:Hermann König
Affiliation:Institut für Angewandte Mathematik, Universität Bonn, Wegelerstrasse 6, 53 Bonn, West Germany
Abstract:Let Ω ? RN be an open set with dist(x, ?Ω) = O(¦ x ¦?l) for x ? Ω and some l > 0 satisfying an additional regularity condition. We give asymptotic estimates for the approximation numbers αn of Sobolev imbeddings
/></figure> over these quasibounded domains Ω. Here <figure class=/></figure> denotes the Sobolev space obtained by completing <math><mtext>C</mtext><msub><mi></mi><mn>0</mn></msub><msup><mi></mi><mn>staggered∞</mn></msup><mtext>(Ω)</mtext></math> under the usual Sobolev norm. We prove <math><mtext>α</mtext><msub><mi></mi><mn>n</mn></msub><mtext>(I</mtext><msub><mi></mi><mn>p,q</mn></msub><msup><mi></mi><mn>m</mn></msup><mtext>) </mtext><mtext>$</mtext><mtext>?</mtext><mtext>n</mtext><msup><mi></mi><mn>?γ</mn></msup></math>, where <figure class=/></figure>. There are quasibounded domains of this type where γ is the exact order of decay, in the case <em>p</em> ? <em>q</em> under the additional assumption that either 1 ? <em>p</em> ? <em>q</em> ? 2 or 2 ? <em>p</em> ? <em>q</em> ? ∞. This generalizes the known results for bounded domains which correspond to <em>l</em> = ∞. Similar results are indicated for the Kolmogorov and Gelfand numbers of <em>I</em><sub><em>p</em>,<em>q</em></sub><sup><em>m</em></sup>. As an application we give the rate of growth of the eigenvalues of certain elliptic differential operators with Dirichlet boundary conditions in <math><mtext>L</mtext><msub><mi></mi><mn>2</mn></msub><mtext>(Ω)</mtext></math>, where Ω is a quasibounded domain of the above type.</td> </tr> <tr> <td align=
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