Rigorous and heuristic treatment of sensitive singular perturbations arising in elliptic shells

TL;DR

This paper investigates singular perturbations in elliptic systems related to thin shell theory, introducing a heuristic boundary reduction method and extending previous work from single equations to systems.

Contribution

It presents a novel approach to analyze systems of PDEs with sensitive singular perturbations, especially in the context of elliptic shells, using heuristic boundary simplifications.

Findings

Extended analysis from single equations to systems of PDEs.

Developed a boundary reduction technique for complex perturbation problems.

Provided insights into the behavior of elliptic shells with boundary conditions not satisfying classical criteria.

Abstract

We consider singular perturbations of elliptic systems depending on a parameter ? such that, for ? = 0 the boundary conditions are not adapted to the equation (they do not satisfy the Shapiro - Lopatinskii condition). The limit holds only in very abstract spaces out of distribu- tion theory involving complexi?cation and non-local phenomena. This system appears in the thin shell theory when the middle surface is el- liptic and the shell is fixed on a part of the boundary and free on the rest. We use a heuristic reasoning applying some simplifications which allow to reduce the original problem in a domain to another problem on its boundary. The novelty of this work is that we consider systems of partial differential equations while in our previous work we were dealing with single equations.