An application of a theorem of G. Zwirner to a class of non-linear elliptic systems in divergence form

TL;DR

This paper applies a theorem by G. Zwirner to find functional solutions of certain non-linear elliptic systems in divergence form, and discusses the existence and uniqueness of these solutions.

Contribution

It introduces a novel application of Zwirner's theorem to a class of coupled non-linear elliptic PDEs with boundary conditions.

Findings

Existence of solutions established under specific conditions.

Uniqueness of solutions demonstrated for the system.

Method provides a new approach to solving coupled elliptic systems.

Abstract

A theorem on the solutions of the problem $U'(w)=\gamma F(U(w),w),\ U(w_1)=u_2,\ U(w_2)=u_2$ is applied for finding the functional solutions of the system of partial differential equations \begin{equation} \nabla\cdot(a(u,w)\nabla u)=0,\ u=u_1\ on \Gamma_1,\ u=u_2\ on \Gamma_2,\ \frac{\partial u}{\partial n}=0\ on\ \Gamma_3 \end{equation} \begin{equation} \nabla\cdot(b(u,w)\nabla w)=0, \ w=w_1\ on \Gamma_1,\ w=w_2\ on\ \Gamma_2,\ \frac{\partial u}{\partial n}=0\ on\ \Gamma_3. \end{equation} The problem of existence and uniqueness of solutions is also considered.