Search results for "Neumann eigenvalue"
showing 6 items of 6 documents
Sharp estimates for eigenfunctions of a Neumann problem
2009
In this paper we provide some bounds for the eigenfunctions of the Laplacian with homogeneous Neumann boundary conditions in a bounded domain Ω of R^n. To this aim we use the so-called symmetrization techniques and the obtained estimates are asymptotically sharp, at least in the bidimensional case, when the isoperimetric constant relative to Ω goes to 0.
Optimal lower bounds for eigenvalues of linear and nonlinear Neumann problems
2013
In this paper we prove a sharp lower bound for the first non-trivial Neumann eigenvalue μ1(Ω) for the p-Laplace operator (p > 1) in a Lipschitz bounded domain Ω in ℝn. Our estimate does not require any convexity assumption on Ω and it involves the best isoperimetric constant relative to Ω. In a suitable class of convex planar domains, our bound turns out to be better than the one provided by the Payne—Weinberger inequality.
The equality case in a Poincaré–Wirtinger type inequality
2016
It is known that, for any convex planar set W, the first non-trivial Neumann eigenvalue μ1 (Ω) of the Hermite operator is greater than or equal to 1. Under the additional assumption that Ω is contained in a strip, we show that β1 (Ω) = 1 if and only if Ω is any strip. The study of the equality case requires, among other things, an asymptotic analysis of the eigenvalues of the Hermite operator in thin domains.
Sharp Poincaré inequalities in a class of non-convex sets
2018
Let $gamma$ be a smooth, non-closed, simple curve whose image is symmetric with respect to the $y$-axis, and let $D$ be a planar domain consisting of the points on one side of $gamma$, within a suitable distance $delta$ of $gamma$. Denote by $mu_1^{odd}(D)$ the smallest nontrivial Neumann eigenvalue having a corresponding eigenfunction that is odd with respect to the $y$-axis. If $gamma$ satisfies some simple geometric conditions, then $mu_1^{odd}(D)$ can be sharply estimated from below in terms of the length of $gamma$ , its curvature, and $delta$. Moreover, we give explicit conditions on $delta$ that ensure $mu_1^{odd}(D)=mu_1(D)$. Finally, we can extend our bound on $mu_1^{odd}(D)$ to a …
An optimal Poincaré-Wirtinger inequality in Gauss space
2013
International audience; Let $\Omega$ be a smooth, convex, unbounded domain of $\mathbb{R}^N$. Denote by $\mu_1(\Omega)$ the first nontrivial Neumann eigenvalue of the Hermite operator in $\Omega$; we prove that $\mu_1(\Omega) \ge 1$. The result is sharp since equality sign is achieved when $\Omega$ is a $N$-dimensional strip. Our estimate can be equivalently viewed as an optimal Poincaré-Wirtinger inequality for functions belonging to the weighted Sobolev space $H^1(\Omega,d\gamma_N)$, where $\gamma_N$ is the $N$% -dimensional Gaussian measure.
A sharp estimate for Neumann eigenvalues of the Laplace-Beltrami operator for domains in a hemisphere
2018
Here, we prove an isoperimetric inequality for the harmonic mean of the first [Formula: see text] non-trivial Neumann eigenvalues of the Laplace–Beltrami operator for domains contained in a hemisphere of [Formula: see text].