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 |λD1,∞(Ω) - λD1,∞ (Br)| = δ1 and |λN1,∞(Ω) - λN1,∞(Br)| = δ2, then there are two balls such that B r\δ1r+1 ⊂ Ω ⊂ B r+δ2r∙/1-δ2r In addition, we obtain a result concerning stability of the Dirichlet ∞-eigenfunctions.