Perturbation method for second order strongly elliptic systems of PDEs with constant coefficients

TL;DR

This paper introduces a perturbation series method for solving second-order strongly elliptic PDE systems with constant coefficients in Jordan domains, demonstrating uniform convergence under specific regularity conditions.

Contribution

It develops a novel perturbation approach for elliptic systems, extending classical solutions to systems with deviations from the Laplacian operator.

Findings

Series converges uniformly in the domain closure

Solution representation as a functional series in powers of the deviation parameter

Requires boundary and boundary function regularity conditions

Abstract

The classical Dirichlet problem for a second-order strongly elliptic system with constant coefficients in a Jordan domain is considered. We show that the solution of the problem can be represented as a functional series in powers of the parameter, which determines the deviation of the system operator from the Laplacian. This series converges uniformly in the closure of the region under the assumption that the boundary of the region and the boundary function satisfy the sufficient regularity conditions: the trace of a conformal mapping of the domain onto a circle composed with the boundary function belongs to the Holder class with exponent greater than 1/2.