Affiliation:
1. Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine
2. Holon Institute of Technology
Abstract
We study the semi-linear Beltrami equation $\omega_{\bar{z}}-\mu(z) \omega_z=\sigma(z)q(\omega(z))$ and show that it is closely related to the corresponding semi-linear equation of the form ${\rm div} A(z)\nabla\,U(z)=G(z) Q(U(z)).$ Applying the theory of completely continuous operators by Ahlfors-Bers and Leray-Schauder, we prove existence of regular solutions both to the semi-linear Beltrami equation and to the given above semi-linear equation in the divergent form, see Theorems 1.1 and 5.2. We also derive their representation through solutions of the semi-linear Vekua type equations and generalized analytic functions with sources. Finally, we apply Theorem 5.2 for several model equations describing physical phenomena in anisotropic and inhomogeneous media.
Funder
The European Federation of Academies of Sciences and Humanities
Publisher
Constructive Mathematical Analysis
Subject
Applied Mathematics,Numerical Analysis,Analysis
Reference34 articles.
1. L. Ahlfors: Lectures on Quasiconformal Mappings, Van Nostrand, New York (1966).
2. L. V. Ahlfors, L. Bers: Riemanns mapping theorem for variable metrics, Ann. Math., 72 (2) (1960), 385–404.
3. K. Astala, T. Iwaniec and G. J. Martin: Elliptic differential equations and quasiconformal mappings in the plane, Princeton Math. Ser., 48, Princeton Univ. Press, Princeton (2009).
4. R. Aris: The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, V. I–II, Clarendon Press, Oxford (1975).
5. 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).
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