Toward the theory of semi-linear Beltrami equations

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.