We study the problem of group classification of quasilinear elliptic equations in a two-dimensional space. The list of all equations of this type admitting solvable Lie algebras of symmetry operators is obtained. Together with the results obtained earlier by the authors, these results give a complete solution of the problem of group classification of quasilinear elliptic equations.
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V. I. Lahno and S. V. Spichak, “Group classification of quasilinear elliptic-type equations. I. Invariance with respect to Lie algebras with nontrivial Levi decomposition,” Ukr. Mat. Zh., 59, No. 11, 1532–1545 (2007); English translation: Ukr. Math. J., 59, No. 11, 1719–1736 (2007).
R. Z. Zhdanov and V. I. Lahno, ”Group classification of heat conductivity equations with a nonlinear source,” J. Phys. A, Math. Gen., 32, 7405–7418 (1999).
P. Basarab-Horwath, V. Lahno, and R. Zhdanov, “The structure of Lie algebras and the classification problem for partial differential equations,” Acta Appl. Math., 69, 43–94 (2001).
V. I. Lahno, S. V. Spichak, and V. I. Stognii, Symmetry Analysis of Equations of Evolution Type [in Russian], Institute of Computer Investigations, Moscow (2004).
G. M. Mubaryakzanov, “On solvable Lie algebras,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 1(32), 114–123 (1963).
G. M. Mubaryakzanov, “Classification of real structures of Lie algebras of order five,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 3(34), 99–106 (1963).
L. V. Ovsyannikov, “Group properties of nonlinear heat conduction equations,” Dokl. Akad. Nauk SSSR, 125, No. 3, 492–495 (1959).
L. V. Ovsyannikov, Group Analysis of Differential Equations [in Russian], Nauka, Moscow (1978).
I. Sh. Akhatov, R. K. Gazizov, and N. Kh. Ibragimov, “Nonlocal symmetries. Heuristic approach,” in: VINITI Series in Contemporary Problems of Mathematics. Recent Advances [in Russian], Vol. 34, VINITI, Moscow (1989), pp. 3–83.
V. I. Fushchich, V. M. Shtelen’, and N. Serov, Symmetry Analysis and Exact Solutions of Nonlinear Equations of Mathematical Physics [in Russian], Naukova Dumka, Kiev (1989).
W. Fushchich, W. Shtelen, and N. Serov, Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics, Kluwer, Dordrecht (1993).
W. I. Fushchych and I. M. Tsyfra, “On a reduction and solutions of the nonlinear wave equations with broken symmetry,” J. Phys. A, Math. Gen., 20, L45–L48 (1987).
E. M. Vorob’ev and N. V. Ignatovich, “Group analysis of a boundary-value problem for equations of a laminar boundary layer,” Mat. Model., 3, No. 11, 116–123 (1991).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 2, pp. 200–215, February, 2011.
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Lahno, V.I., Spichak, S.V. Group classification of quasilinear elliptic-type equations. II. Invariance under solvable Lie algebras. Ukr Math J 63, 236–253 (2011). https://doi.org/10.1007/s11253-011-0501-4
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DOI: https://doi.org/10.1007/s11253-011-0501-4