Magnetostatic levitation and two related linear PDEs in unbounded domains

TL;DR

This paper analyzes a magnetostatic levitation problem leading to a linear PDE in a strip, proving the uniqueness of a particular solution used for force calculations and establishing new uniqueness results for related PDEs in unbounded domains.

Contribution

It demonstrates the uniqueness of a solution used in engineering for magnetostatic levitation and introduces novel uniqueness results for two linear PDEs in unbounded domains.

Findings

Unique solution in a class of bounded regular functions for the PDE in magnetostatic levitation

Validation of the solution's agreement with experimental lift and drag forces

New nonstandard uniqueness results for linear PDEs in unbounded domains

Abstract

We consider a problem occurring in a magnetostatic levitation. The problem leads to a linear PDE in a strip. In engineering literature a particular solution is obtained. Such a solution enables one to compute lift and drag forces of the levitating object. It is in agreement with the experiment. We show that such a solution is unique in a class of bounded regular functions. Moreover, as a byproduct, we obtain nonstandard uniqueness results in two linear PDEs in unbounded domains. One of them is an Eigenvalue problem for the Laplacian in the strip in the nonstandard class of functions.