Localisation of the first eigenfunction of a convex domain. Issue 3 (2nd November 2020)

• Record Type: Journal Article
• Authors: Beck, Thomas
• Is Part Of: Communications in partial differential equations. Volume 46:Issue 3(2021)
• Page Start: 395
• Page End: 412
• Publication Date: 2020-11-02
• Subjects: Convexity -- Dirichlet eigenfunction -- eigenvalue estimates -- level sets; Differential equations, Partial -- Periodicals; 515.353
• DOI: 10.1080/03605302.2020.1843050
• Languages: English
• ISSNs: 0360-5302

Abstract

Abstract: 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 [1, 2 ], imply that the ratio of the L 2 -norm and L ∞ -norm of the eigenfunction is minimised 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 L 2 -norm of the eigenfunction and show that the eigenfunction cannot localise to too small a subset of the domain. As a consequence, we settle a conjecture of van den Berg, [3 ], 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.