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

We prove the existence of an optimal domain for minimizing the buckling load among all, possibly unbounded, open subsets of ${\mathbb{R}}^n$ ($n\ge 2$) 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.