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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, $${}_0F_1$$