Elliptische Randwertprobleme zweiter Ordnung in Gebieten mit einer Fehlstelle. (Elliptic boundary value problems of the second order in domains with a hole)
Let G, \(\Omega \subset R^ n\) be open domains containing the origin with smooth boundaries. Let \(\Omega_{\epsilon}\) consist of points \(\epsilon\) x where \(x\in \Omega\), then \(\Omega_{\epsilon}\subset G\) for all \(\epsilon\) near to zero. The author derives the asymptotics \[ u=w_ 0+v_ 0+\epsilon (w_ 1+v_ 1)+\epsilon^ 2(w_ 2+v_ 2)+... \] for the solution of the problem \[ -\Delta u+c(x)u=f(x)\quad (x\in G-{\bar \Omega}_{\epsilon}),\quad u(x)-f(x)\quad (x\in \partial G),\quad u(x)=0\quad (x\in \partial \Omega_{\epsilon}). \] The summands \(w_ i\) \((v_ i)\) are solutions of certain recurrence of inner (outer) boundary value problems for the same operator \(-\Delta +c(x)\) as above with boundary values given at \(\partial G\) \((\partial \Omega_{\epsilon})\). Similar results are presented for the Neumann conditions \(\partial u/\partial n=0\) on \(\partial \Omega_{\epsilon}\).
scientific article; zbMATH DE number 4136452