On the reverse Faber-Krahn inequalities

Abstract

Payne-Weinberger showed that \textit{`among the class of membranes with given area $A$, free along the interior boundaries and fixed along the outer boundary of given length $L_0$, the annulus ${\Omega }^{\#}$ has the highest fundamental frequency,'} where ${\Omega }^{\#}$ is a concentric annulus with the same area as $\Omega$ and the same outer boundary length as $L_0$. We extend this result for the higher dimensional domains and $p$-Laplacian with $p\in (1,\infty ),$ under the additional assumption that the outer boundary is a sphere. As an application, we prove that the nodal set of the second eigenfunctions of $p$-Laplacian (with mixed boundary conditions) on a ball and a concentric annulus cannot be a concentric sphere.

References in corpus (2)

• Sharp bounds for the first eigenvalue and the torsional rigidity related to some anisotropic operators
• On the strict monotonicity of the first eigenvalue of the $p$-Laplacian on annuli

Cited by in corpus (8)

• A shape variation result via the geometry of eigenfunctions
• Sharp estimates for the first $p$-Laplacian eigenvalue and for the $p$-torsional rigidity on convex sets with holes
• Domain variations of the first eigenvalue via a strict Faber-Krahn type inequality
• Optimal Shapes for the First Dirichlet Eigenvalue of the $p$-Laplacian and Dihedral symmetry
• Szegő-Weinberger type inequalities for symmetric domains with holes
• Reverse Faber-Krahn and Szego-Weinberger type inequalities for annular domains under Robin-Neumann boundary conditions
• Reverse Faber-Krahn inequality for the $p$-Laplacian in Hyperbolic space
• Second-order derivative of domain-dependent functionals along Nehari manifold trajectories