4 problems

Let $N\ge2$, let $\Omega\subset\mathbb R^N$ be a bounded domain, and let $\nu=(N-2)/2$. Denote the Dirichlet eigenvalues of $\Omega$ by…

Dirichlet monotonicity with volume scaling. The function

Let $n\geq 3$. An $n$-gon is a polygon with $n$ sides, and let $\lambda_1$ denote its first Dirichlet eigenvalue. Among all $n$-gons with fixed area, the regular $n$-gon is the pol…

Let $\Omega$ be a bounded plane domain of diameter $D$, and let $0<\lambda_1\leq\lambda_2\leq\cdots$ be the Dirichlet eigenvalues of its Laplacian. Disk eigenvalue-sum conjecture.…