Geodesic
Numerical study of the Zaremba problem
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 500 (2021), pp. 5-9.

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We consider the eigenvalue problem for a two-dimensional Laplace operator with mixed boundary conditions (Zaremba problem), which (presumably) has a smooth solution inside the domain. Calculations show that the operator $-\Delta$ has a negative eigenvalue, i.e., it is not positive definite.
Keywords: numerical algorithms without saturation, Zaremba problem, eigenvalue problem with mixed boundary conditions. 
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S. D. Algazin. Numerical study of the Zaremba problem. Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 500 (2021), pp. 5-9. http://geodesic.mathdoc.fr/item/DANMA_2021_500_a0/

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