3 problems

Triangle eigenvalue bound. The eigenvalue satisfies

Let $\Omega\subseteq\mathbb{R}^2$ be a connected planar domain, and let $u_3$ be an eigenfunction corresponding to a simple third eigenvalue $\lambda_3$ of the Dirichlet Laplacian…

Let $\Omega$ be a bounded domain in Euclidean space $\mathbb{R}^{n}$, and let $\lambda_i$ be the $i$-th Dirichlet eigenvalue of the Laplacian on $\Omega$. Let $\mathbb{B}^{n}$ be a…