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Normalized systems of solutions of the polylinear vector-matrix Lame’s equations. (Russian) Zbl 0973.35048

The system of Lamé’s equations [G. Lamé, Leçons sur la théorie mathématique de l’élasticité des corps solides, Paris, Mallet-Bachelier, 1852] \((E\Delta+\gamma D)u(x)=0\), where \(x\in \mathbb{R}^3\), \(E\) is the unique matrix in \(\mathbb{R}^3\), \(\Delta\) is the Laplace operator, \(\gamma=\text{const}\), and \(D=(\partial x_i \partial x_j)_{\mid i,j= 1,2,3}\) is considered. On the base of quasipolynomial functions \[ F_s^{p,q}(x,f)=\sum_{i=0}^{q-1}(-1)^i\binom{i+p-1}{p-1} \frac{x_1^{2i+2p-2+s}}{(2i+2p-2+s)!}\Delta^if(x_2,x_3) \] the normalized systems of quasipolynomial solutions to the polylinear vector-matrix Lamé’s equations are constructed and investigated.

MSC:

35C05 Solutions to PDEs in closed form
35Q72 Other PDE from mechanics (MSC2000)
35E20 General theory of PDEs and systems of PDEs with constant coefficients
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
35C10 Series solutions to PDEs
33C70 Other hypergeometric functions and integrals in several variables
74B05 Classical linear elasticity
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