The Dirichlet problem for semi-linear equations

We study the Dirichlet problem for the semi–linear partial differential equations $\mathrm{d}\mathrm{i}\mathrm{v}(A\mathrm{\nabla }u)=f(u)$ 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 $C\cap W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,2}(D)$ 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.