Toward the theory of semi-linear Beltrami equations

Authors : Vladimir GUTLYANSKİİ, Olga NESMELOVA, Vladimir RYAZANOV, Eduard YAKUBOV

Pages : 151-163

Doi:10.33205/cma.1248692

View : 46 | Download : 45

Publication Date : 2023-09-15

Article Type : Research Paper

Abstract :We study the semi-linear Beltrami equation $\\omega_{\\bar{z}}-\\muinsert ignore into journalissuearticles values(z); \\omega_z=\\sigmainsert ignore into journalissuearticles values(z);qinsert ignore into journalissuearticles values(\\omegainsert ignore into journalissuearticles values(z););$ and show that it is closely related to the corresponding semi-linear equation of the form ${\\rm div} Ainsert ignore into journalissuearticles values(z);\\nabla\\,Uinsert ignore into journalissuearticles values(z);=Ginsert ignore into journalissuearticles values(z); Qinsert ignore into journalissuearticles values(Uinsert ignore into journalissuearticles values(z););.$ 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.
Keywords : semi linear Beltrami equations, generalized analytic functions with sources, semi linear Poisson type equations, generalized harmonic functions with sources