Uniform stability of the ball with respect to the first Dirichlet and Neumann infinity-eigenvalues

Abstract

In this note we analyze how perturbations of a ball Br ⊂ Rn behaves in terms of their first (non-trivial) Neumann and Dirichlet ∞−eigenvalues when a volume constraint Ln(Ω) = Ln(Br) is imposed. Our main result states that Ω is uniformly close to a ball when it has first Neumann and Dirichlet eigenvalues close to the ones for the ball of the same volume Br. In fact, we show that, if |λ D 1,∞(Ω) − λ D 1,∞(Br)| = δ1 and |λ N 1,∞(Ω) − λ N 1,∞(Br)| = δ2, then there are two balls such that B r δ1r+1 ⊂ Ω ⊂ B r+δ2r 1−δ2r . In addition, we also obtain a result concerning stability of the Dirichlet ∞−eigenfunctions.