Optimal Shape of a Domain which minimizes the first Buckling Eigenvalue

TL;DR

This paper proves the existence of an optimal domain that minimizes the buckling load of a clamped plate, using a penalized variational approach and analyzing the regularity and properties of the solution.

Contribution

It introduces a novel penalized formulation for the shape optimization problem and establishes regularity and qualitative properties of the optimal domain and free boundary.

Findings

Existence of an optimal domain minimizing buckling load.

Regularity of the minimizing function with Lipschitz continuous derivatives.

Qualitative properties of the free boundary.

Abstract

In this paper we prove the existence of an optimal domain which minimizes the buckling load of a clamped plate among all bounded domains with given measure. Instead of treating this variational problem with a volume constraint, we introduce a problem without any constraints, but with a penalty term. We concentrate on the minimizing function and prove that it has Lipschitz continuous first derivatives. Furthermore, we show that the penalized problem and the original problem can be treated as equivalent. Finally, we establish some qualitative properties of the free boundary.