Uniform stability of the ball with respect to the first Dirichlet and Neumann infinity-eigenvalues
Date
2018-01-06
Authors
da Silva, Joao Vitor
Rossi, Julio D.
Salort, Ariel M.
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
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.
Description
Keywords
Infinity-eigenvalues estimates, Infinity-eigenvalue problem, Approximation of domains
Citation
da Silva, J. V., Rossi, J. D., & Salort, A. M. (2018). Uniform stability of the ball with respect to the first Dirichlet and Neumann infinity-eigenvalues. <i>Electronic Journal of Differential Equations, 2018</i>(07), pp. 1-9.
Rights
Attribution 4.0 International