Semi-linear equations and quasiconformal mappings

Abstract

ABSTRACTWe study the Dirichlet problem for the semi-linear partial differential equations in simply connected domains D of the complex plane with continuous boundary data. We prove the existence of the weak solutions u in the class if the Jordan domain D satisfies the quasihyperbolic boundary condition by Gehring–Martio. An example of such a domain that fails to satisfy the standard (A)-condition by Ladyzhenskaya–Ural'tseva and the known outer cone condition is given. We also extend our results to simply connected non-Jordan domains formulated in terms of the prime ends by Carathéodory. Our approach is based on the theory of the logarithmic potential, singular integrals, the Leray–Schauder technique and a factorization theorem in Gutlyanskii et al. [On quasiconformal maps and semi-linear equations in the plane. Ukr Mat Visn. 2017;14(2):161–191]. This theorem allows us to represent u in the form where stands for a quasiconformal mapping of D onto the unit disk , generated by the measurable matrix function and U is a solution of the corresponding quasilinear Poisson equation in the unit disk . In the end, we give some applications of these results to various processes of diffusion and absorption in anisotropic and inhomogeneous media.