The Dirichlet problem for semi-linear equations

Abstract

We study the Dirichlet problem for the semi--linear partial differential equations ${\rm div}\,(A\nabla u)=f(u)$ in simply connected domains $D$ of the complex plane $\mathbb C$ with continuous boundary data. We prove the existence of the weak solutions $u$ in the class $C\cap W^{1,2}_{\rm loc}(D)$ if a 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 Caratheodory. Our approach is based on the theory of the logarithmic potential, singular integrals, the Leray--Schauder technique and a factorization theorem in \cite{GNR2017}. This theorem allows us to represent $u$ in the form $u=U\circ\omega,$ where $\omega(z)$ stands for a quasiconformal mapping of $D$ onto the unit disk ${\mathbb D}$, generated by the measurable matrix function $A(z),$ and $U$ is a solution of the corresponding quasilinear Poisson equation in the unit disk ${\mathbb D}$. In the end, we give some applications of these results to various processes of diffusion and absorption in anisotropic and inhomogeneous media.