Geodesic
The Dirichlet Problem for Higher-Order Partial Differential Equations
Matematičeskie zametki, Tome 97 (2015) no. 2, pp. 262-276.

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For higher-order partial differential equations in two or three variables, the Dirichlet problem in rectangular domains is studied. Small denominators hampering the convergence of series appear in the process of constructing the solution of the problem by the spectral decomposition method. A uniqueness criterion for the solution is established. In the two-dimensional case, estimates justifying the existence of a solution of the Dirichlet problem are obtained. In the three-dimensional case where the domain is a cube, it is shown that the uniqueness of the solution of the Dirichlet problem is equivalent to the great Fermat problem.
Keywords: higher-order partial differential equation, Dirichlet problem, spectral decomposition method, Fourier series, Fermat problem. 
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K. B. Sabitov. The Dirichlet Problem for Higher-Order Partial Differential Equations. Matematičeskie zametki, Tome 97 (2015) no. 2, pp. 262-276. http://geodesic.mathdoc.fr/item/MZM_2015_97_2_a8/

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