Abstract
We study the Dirichlet problem with continuous boundary data in simply connected domains D of the complex plane for the semi-linear partial differential equations whose linear part has the divergent form. We prove that if a Jordan domain D satisfies the so-called quasihyperbolic boundary condition, then the problem has regular (continuous) weak solutions whose first generalized derivatives by Sobolev are integrable in the second degree. We give a suitable example of a Jordan domain with the quasihyperbolic boundary condition that fails to satisfy both the well-known (A)-condition and the outer cone condition. We also extend these results to some non-Jordan domains in terms of the prime ends by Caratheodory. The proofs are based on our factorization theorem established earlier. This theorem allows us to represent solutions of the semilinear equations in the form of composition of solutions of the corresponding quasilinear Poisson equation in the unit disk and quasiconformal mapping of D onto the unit disk generated by the measurable matrix function of coefficients. In the end we give applications to relevant problems of mathematical physics in anisotropic inhomogeneous media.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L. V. Ahlfors, Lectures on Quasiconformal Mappings, Vol. 10 of Van Nostrand Mathematical Studies (Van Nostrand, Toronto, New York, London, 1966).
K. Astala, T. Iwaniec, and G. J. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Vol. 48 of Princeton Mathematical Series (Princeton Univ. Press, Princeton, 2009).
K. Astala and P. Koskela, “Quasiconformal mappings and global integrability of the derivative,” J. Anal. Math. 57, 203–220 (1991).
R. Aris, The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts (Clarendon, Oxford, 1975), Vols. 1, 2.
M. G. Arsove, “The Osgood–Taylor–Caratheodory theorem,” Proc. Am. Math. Soc. 19, 38–44 (1968).
G. I. Barenblatt, Ja. B. Zel’dovic, V. B. Librovich, and G. M. Mahviladze, The Mathematical Theory of Combustion and Explosions (Consult. Bureau, New York, 1985).
J. Becker and Ch. Pommerenke, “Hölder continuity of conformal mappings and nonquasiconformal Jordan curves,” Comment. Math. Helv. 57, 221–225 (1982).
B. V. Bojarski, “Homeomorphic solutions of Beltrami systems,” Dokl. Akad. Nauk SSSR 102, 661–664 (1955).
B. V. Bojarski, “Generalized solutions of a system of differential equations of the first order and elliptic type with discontinuous coefficients,” Mat. Sb. 43 (85), 451–503 (1957).
B. Bojarski, V. Gutlyanskii, O. Martio, and V. Ryazanov, Infinitesimal Geometry of Quasiconformal and Bi-Lipschitz Mappings in the Plane, Vol. 19 of EMS Tracts in Mathematics (Eur. Math. Soc., Zürich, 2013).
M. Borsuk and V. Kondratiev, Elliptic Boundary Value Problems of Second Order in Piecewise Smooth Domains, Vol. 69 of North-Holland Mathematical Library (Elsevier Science, Amsterdam, 2006).
C. Caratheodory, “Uber die gegenseitige Beziehung der Rander bei der konformen Abbildungen des Inneren einer Jordanschen Kurve auf einen Kreis,” Math. Ann. 73, 305–320 (1913).
C. Caratheodory, “Über die Begrenzung der einfachzusammenhängenderGebiete,” Math. Ann. 73, 323–370 (1913).
E. F. Collingwood and A. J. Lohwator, The Theory of Cluster Sets, Vol. 56 of Cambridge Tracts in Mathematics and Mathematical Physics (Cambridge Univ. Press, Cambridge, 1966).
J. I. Diaz, Nonlinear partial differential equations and free boundaries, Vol. 1: Elliptic Equations, Vol. 106 of Research Notes inMathematics (Pitman, Boston, 1985).
F. W. Gehring, Characteristic Properties of Quasidisks, Vol. 84 of Seminaire de Mathematiques Superieures (Press. de Univ. Montreal,Montreal, Quebeck, 1982).
F. W. Gehring and K. Hag, The Ubiquitous Quasidisk, Vol. 184 of Mathematical Surveys and Monographs (Am. Math. Soc., Providence, RI, 2012).
F. W. Gehring and O. Martio, “Quasiextremal distance domains and extension of quasiconformal mappings,” J. AnalyseMath. 45, 181–206 (1985).
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Vol. 224 of Grundlehren der Mathematischen Wissenschaften (Springer, Berlin, 1983).
V. Ya. Gutlyanskii, O. V. Nesmelova, and V. I. Ryazanov, “On quasiconformal maps and semilinear equations in the plane,” J. Math. Sci. (US) 229, 7–29 (2018).
V. Ya. Gutlyanskii, O. V. Nesmelova, and V. I. Ryazanov, “On the Dirichlet problem for quasilinear Poisson equations,” Proc. Inst. Appl. Math. Mech., NAS Ukr. 31, 28–37 (2017).
V. Ya. Gutlyanskii and V. I. Ryazanov, The Geometric and Topological Theory of Functions and Mappings (Naukova Dumka, Kiev, 2011) [in Russian].
V. Gutlyanskii, V. Ryazanov, U. Srebro, and E. Yakubov, The Beltrami Equation: A Geometric Approach, Vol. 26 of Developments in Mathematics (Springer, New York etc., 2012).
I. Kuzin and S. Pohozaev, Entire Solutions of Semi-linear Elliptic Equations, Vol. 33 of Progress in Nonlinear Differential Equations and their Applications (Birkhäuser, Basel, 1997).
E. M. Landis, Second Order Equations of Elliptic and Parabolic Type, Vol. 171 of Translations of Mathematical Monographs (Am. Math. Soc., Providence, RI, 1998).
O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations (Academic, New York, London, 1968).
O. Lehto and K. I. Virtanen, QuasiconformalMappings in the Plane, 2nd ed. (Springer, Berlin,Heidelberg, New York, 1973).
M. Marcus and L. Veron, Nonlinear Second Order Elliptic Equations Involving Measures, Vol. 21 of De Gruyter Series in Nonlinear Analysis and Applications (De Gruyter, Berlin, 2014).
O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, Moduli in Modern Mapping Theory (Springer, New York, 2009).
R. M. Nasyrov and S. R. Nasyrov, “Convergence of S. A. Khristianovich’s approximate method for solving the Dirichlet problem for an elliptic equation,” Dokl. Akad. Nauk SSSR 291, 294–298 (1986) [in Russian].
W. Osgood and E. Taylor, “Conformal transformations on the boundaries of their regions of definition,” Trans. Am. Math. Soc. 14, 277–298 (1913).
S. I. Pokhozhaev, “On an equation of combustion theory,” Math. Notes 88, 48–56 (2010).
M. Vuorinen, Conformal Geometry of Quasiregular Mappings, Lecture NotesMath. 1319 (1988).
S. E. Warschawski, “On differentiability at the boundary in conformal mapping,” Proc. Am. Math. Soc. 12, 614–620 (1961).
Author information
Authors and Affiliations
Corresponding author
Additional information
(Submitted by A. M. Elizarov)
Rights and permissions
About this article
Cite this article
Gutlyanskiĭ, V., Ryazanov, V. Quasiconformal Mappings in the Theory of Semi-linear Equations. Lobachevskii J Math 39, 1343–1352 (2018). https://doi.org/10.1134/S1995080218090251
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080218090251