The Dirichlet problem for semi-linear equations

TL;DR

This paper proves the existence of weak solutions for semi-linear PDEs with boundary data in certain complex domains, using quasiconformal mappings and potential theory, with applications to diffusion processes.

Contribution

It extends existence results to complex domains satisfying quasihyperbolic boundary conditions, including non-Jordan domains, using a novel factorization approach involving quasiconformal maps.

Findings

Existence of weak solutions in complex domains with quasihyperbolic boundary conditions.

Representation of solutions via quasiconformal mappings and solutions of quasilinear equations.

Applications to diffusion and absorption in anisotropic media.

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.