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Ill-posed boundary value problems for the \(n\)-th order polyharmonic equation. (English. Russian original) Zbl 0842.35026
Russ. Math. 38, No. 10, 16-21 (1994); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1994, No. 10 (389), 19-25 (1994).
First, for the Laplace operator \(\Delta\) there are solutions of \(\Delta^3u + c_1 \Delta^2u + c_2 \Delta u + c_3u = 0\) represented by modified Bessel functions. Boundary conditions are \(u |_{\Gamma_k} = \varphi_k\) on the concentric circles \((k = 1,2,3)\). If the roots of \(\lambda^3 + c_1 \lambda^2 + c_2 \lambda + c_3 = 0\) are nonnegative and distinct and the Fourier coefficients of the \(\varphi_k\) \((k = 1,2)\) satisfy certain inequalities and \(\varphi_3 \in C\), then there exists a unique solution in the class of uniformly bounded analytic functions.
Second, for the Laplace-Beltrami operator (constant curvature) solutions \(u(z, \varphi)\) of \(L^3u + c_1 L^2u + c_2 Lu + c_3 u = 0\) on the spherical space \(P^2\) are represented by Legendre functions. If the \(\Gamma_i\) in \(P^2\) are given by \(z = \cos \theta = R_i\) \((i = 1,2,3)\), \(0 \leq \theta < \tau\) and the \(R_i\) and \(\lambda_i\) satisfy the conditions of the first case and the Fourier coefficients of the \(\varphi_\ell\) \((\ell = 1,2)\) satisfy certain inequalities, then there exists a unique solution in the class of uniformly bounded analytic functions. Here the reviewer looked for conditions for \(\varphi_3\).
MSC:
35J40 Boundary value problems for higher-order elliptic equations
35C05 Solutions to PDEs in closed form
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
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