A shape variation result via the geometry of eigenfunctions

Abstract

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. These fine geometric properties, together with the shape calculus, help us to prove that the first eigenvalue is strictly decreasing as the inner ball moves towards the boundary of the outer ball.

21 pages, 3 figures

References in corpus (2)

• On the symmetry of minimizers
• The shape of extremal functions for Poincaré-Sobolev-type inequalities in a ball

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• Strict Faber-Krahn type inequality for the mixed local-nonlocal operator under polarization
• A sharp bound for the first Robin-Dirichlet eigenvalue
• Approximation of the first Steklov-Dirichlet eigenvalue on eccentric spherical shells in general dimensions