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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     author = {S. D. Algazin},
     title = {Numerical study of the {Zaremba} problem},
     journal = {Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleni\^a},
     pages = {5--9},
     publisher = {mathdoc},
     volume = {500},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DANMA_2021_500_a0/}
}
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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/