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. 
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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