Integral approach to sensitive singular perturbations

TL;DR

This paper investigates singular perturbation elliptic problems with boundary conditions that are incompatible at the limit, providing an elementary model and an integral method to understand the solutions as the perturbation parameter approaches zero.

Contribution

It introduces an integral approach to analyze sensitive singular perturbations with incompatible boundary conditions, highlighting the limit behavior and solution structure.

Findings

Elementary model illustrating the limit process

Heuristic integral procedure for small perturbations

Application to thin shell theory with elliptic surfaces

Abstract

We consider singular perturbation elliptic problems 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 only holds in very abstract spaces out of distribution theory involving complexi?cation and non-local phenomena. We give a very elementary model problem showing the main features of the limit process, as well as a heuristic integral procedure for obtain- ing a description of the solutions for small ?. Such kind of problems appear in thin shell theory when the middle surface is elliptic and the shell is fixed by a part of the boundary and free by the rest.