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Estimates for the conditional stability of a two-contour boundary value problem in the disk. (English. Russian original) Zbl 0961.35040
Russ. Math. 42, No. 10, 5-12 (1998); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1998, No. 10, 7-14 (1998).
From the introduction: We show that the problem \[ \Delta^n u+ c_1\Delta^{n- 1}u+\cdots+ c_nu= 0, \] \[ u(x)|_{\Gamma_1}= \varphi_1(x),\dots, u(x)|_{\Gamma_n}= \varphi_n(x), \] where the closed contours \(\Gamma_k\) bound the domains \(D_k\); moreover, \(D_1\subset D_2\subset\cdots\subset D_n\subseteq D\), is ill-posed. An important role is played by the conditional stability of the problem. In particular, we consider the conditional stability for \(n= 2\), i.e., we consider the fourth-order equation \[ K_4 u=\Delta^2 u+ c_1\Delta u+ c_2 u=0, \] when the contours \(\Gamma_1\) and \(\Gamma_2\) have the form of concentric circumferences with the radii \(R_1\) and \(R_2\), respectively, \(R_2> R_1\).
MSC:
35J40 Boundary value problems for higher-order elliptic equations
30E25 Boundary value problems in the complex plane
35B35 Stability in context of PDEs
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