Semilinear equations in the plane with measurable data

1Gutlyanskii, VYa., 1Nesmelova, OV, 1Ryazanov, VI
1Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Sloviansk
Dopov. Nac. akad. nauk Ukr. 2018, 2:12-18
Section: Mathematics
Language: English

We study semilinear partial differential equations in the plane, the linear part of which is written in a divergence form. The main result is given as a factorization theorem. This theorem states that every weak solution of such an equation can be represented as a composition of a weak solution of the corresponding isotropic equation in a canonical domain and a quasiconformal mapping agreed with a matrix-valued measurable coefficient appearing in the divergence part of the equation. The latter makes it possible, in particular, to remove the regularity restrictions on the boundary in the study of boundary-value problems for such semilinear equations.

Keywords: Beltrami equation, quasiconformal mappings, semilinear elliptic equations
  1. Ahlfors, L. V. (1966). Lectures on quasiconformal mappings. Princeton, N.J.: Van Nostrand. Reprinted by Wadsworth Ink. Belmont, 1987.
  2. Lehto, O. & Virtanen, K. I. (1973). Quasiconformal mappings in the plane. 2nd ed. Berlin, Heidelberg, New York: Springer. doi:
  3. Gutlyanskii, V., Ryazanov, V., Srebro, U. & Yakubov, E. (2012). The Beltrami Equation: A Geometric Approach. Developments in Mathematics. Vol. 26. New York: Springer. doi:
  4. Gutlyanskii, V., Nesmelova, O. & Ryazanov, V. (2016). On a model semilinear elliptic equation in the plane. Ukr. Mat. Visn., 13, No. 1, pp. 91-105; J. Math. Sci., 2017, 220, No. 5, pp. 603-614.
  5. Gutlyanskii, V. Ya., Nesmelova, O. V. & Ryazanov, V. I. (2017). Semilinear equations in a plane and quasiconformal mappings. Dopov. Nac. akad. nauk Ukr., No. 1, pp. 10-16. doi:
  6. Astala, K., Iwaniec, T. & Martin, G. (2009). Elliptic partial differential equations and quasiconformal mappings in the plane. Princeton Mathematical Series. Vol. 48. Princeton, N.J.: Princeton Univ. Press.
  7. Gol'dshtein, V. & Ukhlov, A. (2010). About homeomorphisms that induce composition operators on Sobolev spaces. Complex Var. Elliptic Equ., 55, No. 8-10, pp. 833-845. doi:
  8. Ukhlov, A. (1993). Mappings that generate embeddings of Sobolev spaces. Sibirsk. Mat. Zh., 34, No. 1, pp. 185-192 (in Russian); Siberian Math. J., 34, No.1, pp. 165-171. doi:
  9. Vodopyanov, S. K. & Ukhlov, A. (1998). Sobolev spaces and (P,Q)-quasiconformal mappings of Carnot groups. Sib. Mat. Zh., 39, No. 4, pp. 665-682 (in Russian); Math. J., 39, No. 4, pp. 665-682. doi:
  10. Sobolev, S. L. (1941). On some transformation groups of an n-dimensional space. Dokl. AN SSSR, 32, No. 6, pp. 380-382 (in Russian).
  11. Gol'dshtein, V., Gurov, L. & Romanov, A. (1995). Homeomorphisms that induce monomorphisms of Sobolev spaces. Israel J. Math., 91, No. 1-3, pp. 31-60. doi:
  12. Vodop'yanov, S. K. (2012). On the regularity of mappings inverse to the Sobolev mapping. Mat. Sb., 203, No. 10, pp. 3-32 (in Russian); Sb. Math., 203, No. 9-10, pp. 1383-1410. doi:
  13. Vodop'yanov, S. K. & Evseev, N. A. (2014). Isomorphisms of Sobolev spaces on Carnot groups and quasi-isometric mappings. Sibirsk. Mat. Zh., 55, No. 5, pp. 1001-1039 (in Russian); Math. J., 55, No. 5, pp. 817-848. doi:
  14. Bojarski, B., Gutlyanskii, V., Martio, O. & Ryazanov, V. (2013). Infinitesimal geometry of quasiconformal and bi-lipschitz mappings in the plane. EMS Tracts in Mathematics. Vol. 19. Zurich: European Mathematical Society. doi:
  15. Goluzin, G.M. (1969). Geometric Theory of functions of a complex variable. Providence, Rhode Island: Amer. Math. Soc.