On the optimal domain for minimizing the buckling load of a clamped plate

TL;DR

This paper proves the existence of an optimal domain that minimizes the buckling load of a clamped plate among all open subsets with a fixed measure, extending previous 2D results to higher dimensions.

Contribution

It extends the existence results for optimal domains minimizing buckling load from 2D to higher dimensions with possibly unbounded sets.

Findings

Existence of an optimal domain for buckling load minimization proven in higher dimensions.

The approach extends previous 2D results to n-dimensional spaces.

Method based on eigenfunction focus and measure constraints.

Abstract

We prove the existence of an optimal domain for minimizing the buckling load among all, possibly unbounded, open subsets of $\mathbb{R}^n$ ($n\geq 2$) with given measure. Our approach is based on the extension of a 2-dimensional existence result of Ashbaugh and Bucur and on the idea of Alt and Caffarelli to focus on the eigenfunction.