Geodesic
On solutions with power-law singularities of the homogeneous Dirichlet problem for the Laplace equation in domains with biquadratic boundaries
Contemporary Mathematics. Fundamental Directions, Proceedings of the Fifth International Conference on Differential and Functional-Differential Equations (Moscow, August 17–24, 2008). Part 1, Tome 35 (2010), pp. 33-43.

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Properties of the Dirichlet problem for the Laplace equation in a bounded plane domain of a special type are studied in a certain class of solutions with power-law singularities. We prove that if a harmonic function is allowed to have a finite number of poles, then it can satisfy the trivial Dirichlet condition on certain curves of the studied family. The specified curves are selected, and it is shown that the set of those curves is dense (in a certain sense) in the studied family. 
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V. P. Burskii; A. A. Telitsyna. On solutions with power-law singularities of the homogeneous Dirichlet problem for the Laplace equation in domains with biquadratic boundaries. Contemporary Mathematics. Fundamental Directions, Proceedings of the Fifth International Conference on Differential and Functional-Differential Equations (Moscow, August 17–24, 2008). Part 1, Tome 35 (2010), pp. 33-43. http://geodesic.mathdoc.fr/item/CMFD_2010_35_a2/

[1] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii. Gipergeometricheskaya funktsiya. Funktsii Lezhandra, Nauka, M., 1965

[2] Burskii V. P., Zhedanov A. S., “O zadache Dirikhle dlya uravneniya kolebaniya struny, probleme Ponsele, uravnenii Pellya–Abelya i nekotorykh drugikh svyazannykh s nimi zadachakh”, Ukr. mat. zhurn., 58 (2006), 435–450 | MR

[3] Egorov Yu. V., Shubin M. A., “Lineinye differentsialnye uravneniya s chastnymi proizvodnymi. Osnovy klassicheskoi teorii”, Itogi nauki i tekhniki. Sovremennye problumy matematiki. Fundamentalnye napravleniya, 30, VINITI, 1988, 5–255 | MR | Zbl

[4] Telitsyna A. A., “Zadacha Dirikhle dlya volnovogo uravneniya v oblasti, ogranichennoi bikvadratnoi krivoi”, Trudy IPMM NAN Ukrainy, 13, 2006, 198–210 | MR | Zbl

[5] John F., “The Dirichlet problem for a hyperbolic equation”, Amer. J. Math., 63 (1941), 141–154 | DOI | MR | Zbl

[6] Iatrou A., Roberts J. A. G., “Integrable mappings of the plane preserving biquadratic invariant curves II”, Nonlinearity, 15 (2002), 459–489 | DOI | MR | Zbl

[7] Veron L., Singularities of solutions of second order quasilinear equations, Longman, Harlow, 1996 | MR | Zbl