Calculus of Variations and Geometric Measure Theory

T. V. ANOOP - K. ASHOK KUMAR - S. Kesavan

A shape variation result via the geometry of eigenfunctions

created by k on 28 Oct 2020
modified on 25 Sep 2021


Published Paper

Inserted: 28 oct 2020
Last Updated: 25 sep 2021

Journal: Journal of Differential Equations
Volume: 298
Pages: 430-462
Year: 2021

ArXiv: 2009.13967 PDF
Links: Link


We discuss some of the geometric properties, such as the foliated Schwarz symmetry, the monotonicity along the axial and the affine-radial directions, of the first eigenfunctions of the Zaremba problem for the Laplace operator on annular domains. Together with the shape calculus, these fine geometric properties help us to prove that the first eigenvalue is strictly decreasing as the inner ball moves towards the boundary of the outer ball.

Keywords: shape derivative, Torsional rigidity, Geometry of first eigenfunction, Foliated Schwarz symmetry, Monotonicity of first eigenvalue, Zaremba problem