The perturbation method for the skew-symmetric strongly elliptic systems of PDEs

TL;DR

This paper introduces a perturbation method to solve skew-symmetric strongly elliptic PDE systems by expressing solutions as convergent power series, simplifying the analysis by relating them to Laplace and Poisson equations.

Contribution

The paper develops a novel perturbation approach for skew-symmetric elliptic systems, linking solutions to classical Laplace and Poisson problems, and proves convergence of the series.

Findings

Series converges uniformly in the domain

Solutions can be constructed via sequential Dirichlet problems

Method simplifies solving complex elliptic systems

Abstract

For a Jordan domain with sufficiently smooth boundaries, the solution to the Dirichlet problem for second order skew-symmetric strongly elliptic system with constant coefficients and regular enough boundary data is constructed in the form of a power series of a small parameter describing the perturbation of the given system from the Laplace one. The coefficients of this series are the functions that are determined sequentially as solutions to special Dirichlet problems for the usual Laplace and Poisson equations. The obtained series converges uniformly in the closure of the domain under consideration.