2019/10/10 by Thomas Beck, Beck, Thomas · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1910.04905
openalex publication_date 2019/10/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the first Dirichlet eigenfunction of the Laplacian in a n-dimensional convex domain. For domains of a fixed inner radius, estimates of Chiti imply that the ratio of the L2-norm and L∞-norm of the eigenfunction is minimized when the domain is a ball. However, when the eccentricity of the domain is large the eigenfunction should spread out at a certain scale and this ratio should increase. We make this precise by obtaining a lower bound on the L2-norm of the eigenfunction and show that the eigenfunction cannot localize to too small a subset of the domain. As a consequence, we settle a conjecture of van den Berg, in the general n-dimensional case. The main feature of the proof is to obtain sufficiently sharp estimates on the first eigenvalue in order to estimate the first derivatives of the eigenfunction.