On the plane Lam\'e-Navier system in fractal domains

TL;DR

This paper develops a method to explicitly solve the two-dimensional Lamé-Navier system in plane elasticity, including regions with fractal boundaries, by reformulating it with Cauchy-Riemann operators and introducing a generalized Teodorescu operator.

Contribution

It introduces a new approach using Cauchy-Riemann operators and a generalized Teodorescu operator to solve the Lamé-Navier system in complex fractal domains.

Findings

Explicit solutions for the Lamé-Navier system in fractal domains.

Reformulation of the system using Cauchy-Riemann operators.

Introduction of a generalized Teodorescu operator for wide classes of regions.

Abstract

This paper is devoted to study a fundamental system of equations in plane Linear Elasticity Theory, the two-dimensional Lam\'e-Navier system. We rewrite them in a compressed form in terms of the Cauchy-Riemann operators and it allows us to solve a kind of Riemann problem for this system. A generalized Teodorescu operator, to be introduced here, provides the means for obtaining the explicit solution of this problem for a very wide classes of regions, including those with a fractal boundary.