3 problems

Let $\Omega \subset \mathbb{R}^d$ be a bounded domain with regular boundary, and let $(\varphi_n)_{n \geq 1}$ be an $L^2$-orthonormal basis of Dirichlet-Laplacian eigenfunctions. F…

Let $\Omega\subset\mathbb R^n$ be a convex domain, and let $u$ be its ground state eigenfunction for the Dirichlet Laplacian, normalized in $L^\infty$. Ishige–Salani–Takatsu's conj…

The 1/2-log-concavity conjecture. If $\phi\leq 1$ in $\Omega$, then $\phi$ is $1/2$-log-concave in $\Omega$.