4 problems

Equality conjecture. It is conjectured that

Hausdorff-measure lower-bound conjecture. There exists a constant $C>0$, depending on $n$, $k$, $hR$, and an upper bound for $\mathscr{H}^k(E)$, such that

Let $0Omega$ be the unit disk, let $0betageq 0$, and let E^ be an optimal set for Problem … when $\Omega$ is the unit disk and $\beta\geq 0$. The statement is motivated by nume…

Consider the principal-eigenvalue shape-optimization problem in dimension $2$ with the framework and parameters described in the paper, and let $E^*$ denote a minimizing set. A con…