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- W2613871750 abstract "In this note we analyze how perturbations of a ball $mathfrak{B}_r subset mathbb{R}^n$ behaves in terms of their first (non-trivial) Neumann and Dirichlet $infty-$eigenvalues when a volume constraint $mathscr{L}^n(Omega) = mathscr{L}^n(mathfrak{B}_r)$ is imposed. Our main result states that $Omega$ 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 $mathfrak{B}_r$. In fact, we show that, if $$ |lambda_{1,infty}^D(Omega) - lambda_{1,infty}^D(mathfrak{B}_r)| = delta_1 quad text{and} quad |lambda_{1,infty}^N(Omega) - lambda_{1,infty}^N(mathfrak{B}_r)| = delta_2, $$ then there are two balls such that $$mathfrak{B}_{frac{r}{delta_1 r+1}} subset Omega subset mathfrak{B}_{frac{r+delta_2 r}{1-delta_2 r}}.$$ In addition, we also obtain a result concerning stability of the Dirichlet $infty-$eigen-functions." @default.
- W2613871750 created "2017-05-19" @default.
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- W2613871750 date "2017-05-08" @default.
- W2613871750 modified "2023-09-27" @default.
- W2613871750 title "Uniform stability of the ball with respect to the first Dirichlet and Neumann $infty-$eigenvalues" @default.
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