A Constructive Approach to a Class of Overdetermined Problems

TL;DR

This paper presents a constructive method for solving a class of overdetermined problems related to the torsion problem, using asymptotic series and Nekhoroshev Theorem tools, with practical approximations demonstrated through an example.

Contribution

It introduces a fully constructive approach to perturbed overdetermined problems, establishing the existence of infinitely many highly accurate approximants.

Findings

Existence of infinitely many solutions with negligible approximation error

Application of asymptotic series and Nekhoroshev Theorem tools

Demonstration of the constructive approach via an example

Abstract

In this paper we study an overdetermined problem which is directly related to the well known torsion problem studied by J. Serrin. A perturbed version of the latter is tackled by using asymptotic series as well as tools borrowed from the celebrated Nekhoroshev Theorem. In a similar fashion to this class of results, we establish the existence of infinitely many approximants for the perturbed problem's solution, whose approximation error is so small which can be regarded as negligible for practical applications. The approach is fully constructive and this feature is demonstrated via an example in the final section.