1Gutlyanskii, VYa. 1Nesmelova, OV 1Ryazanov, VI 1Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Sloviansk |
Dopov. Nac. akad. nauk Ukr. 2018, 4:9-15 |
https://doi.org/10.15407/dopovidi2018.04.009 |
Section: Mathematics |
Language: English |
Abstract: Presented by Corresponding Member of the NAS of Ukraine V.Ya. Gutlyanskii We study semilinear elliptic equations of the form $div(A(z)\nabla u) = f (u)$ in $\Omega \subset \mathbb{C}$, where $A(z)$ stands for a sym metric $2×2$ matrix function with measurable entries, $detA =1$, and such that $1/ K |\xi|^{2} \leqslant \left \langle A(z)\xi, \xi \right \rangle\leqslant K|\xi|^{2}, \xi \in R^{2}, 1\leqslant K < \infty$. Making use of our Factorization theorem, we give some explicit solutions for the above equation if $f = e^{u}$ or $f = u^{q}$, when matrices $A(z)$ are chosen in an appropriate form.
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Keywords: blowup solutions, quasiconformal mappings, semilinear PDE |
References:
- Ahlfors, L. V. (1966). Lectures on quasiconformal mappings. Princeton, N.J.: Van Nostrand.
- Astala, K., Iwaniec, T. & Martin, G. (2009). Elliptic partial differential equations and quasiconformal mappings in the plane. Princeton Mathematical Series. (Vol. 48). Princeton, NJ: Princeton University Press.
- Bandle, C. & Marcus, M. (1995). Asymptotic behavior of solutions and their derivatives, for semilinear elliptic problems with blowup on the boundary. Ann. Inst. H. Poincaré. Anal. Non Lineaire, 12, No. 2, pp. 155-171. doi: https://doi.org/10.1016/S0294-1449(16)30162-7
- Bojarski, B., Gutlyanskii, V., Martio, O. & Ryazanov, V. (2013). Infinitesimal geometry of quasiconformal and bilipschitz mappings in the plane. EMS Tracts in Mathematics. (Vol. 19). Zurich: European Mathematical Society. doi: https://doi.org/10.4171/122
- Gutlyanskii, V., Ryazanov, V., Srebro, U. & Yakubov, E. (2012). The Beltrami Equation: A Geometric Approach. Developments in Mathematics. (Vol. 26). New York: Springer. doi: https://doi.org/10.1007/978-1-4614-3191-6
- Lehto, O. & Virtanen, K. I. (1973). Quasiconformal mappings in the plane. 2nd ed. Berlin, Heidelberg, New York: Springer. doi: https://doi.org/10.1007/978-3-642-65513-5
- Gutlyanskii, V., Nesmelova, O. & Ryazanov, V. (2017). On quasiconformal maps and semilinear equations in the plane. Ukr. Mat. Visn., 14, No. 2, pp. 161-191; J. Math. Sci., 229, No. 1, pp. 729.
- Gutlyanskii, V. Ya., Nesmelova, O. V. & Ryazanov, V. I. (2017). Semilinear equations in a plane and quasiconformal mapping. Dopov. Nac. akad. nauk Ukr., No. 1, pp. 10-16. doi: https://doi.org/10.15407/dopovidi2017.01.010
- Diaz, J. I. (1985). Nonlinear partial differential equations and free boundaries, Vol. I, Elliptic equations. Research Notes in Mathematics (Vol. 106). Boston: Pitman.
- Marcus, M. & Veron, L. (2014). Nonlinear second order elliptic equations involving measures. De Gruyter Series in nonlinear analysis and applications. (Vol. 21). Berlin, Boston: Walter de Gruyter.
- Bieberbach, L. (1916). $\Delta u = e^{u}$ und die automorphen Funktionen. Math. Ann., 77, pp. 173-212. doi: https://doi.org/10.1007/BF01456901
- Gutlyanskii, V.Ya. & Ryazanov, V. I. (1995). On the theory of the local behavior of quasiconformal mappings. Izv. Ross. akad. nauk. Ser. Mat., 59, No. 3, pp. 31-58 (in Russian); Izv. Math., 59, No. 3, pp. 471-498. Received 07.12.2017