**SMS scnews item created by Daniel Daners at Fri 23 Mar 2018 1614**

Type: Seminar

Distribution: World

Expiry: 27 Mar 2018

**Calendar1: 27 Mar 2018**

**CalLoc1: AGR Carslaw 829**

CalTitle1: PDE Seminar: Sharp bounds for Neumann eigenvalues (Brandolini)

Auth: daners@dora.maths.usyd.edu.au

### PDE Seminar

# Sharp bounds for Neumann eigenvalues

### Brandolini

Barbara Brandolini

Università degli Studi di Napoli “Federico II”, Italy

Tue 27th Mar 2018, 2-3pm, Carslaw Room 829 (AGR)

## Abstract

We prove a sharp lower bound for the first nontrivial Neumann eigenvalue
${\mu}_{1}\left(\Omega \right)$ of the
$p$-Laplace operator
($p>1$) in a Lipschitz,
bounded domain $\Omega $
in ${\mathbb{R}}^{n}$.
Differently from the pioneering estimate by Payne-Weinberger,
our lower bound does not require any convexity assumption on
$\Omega $,
it involves the best isoperimetric constant relative to
$\Omega $ and it is sharp, at least
when $p=n=2$, as the isoperimetric
constant relative to $\Omega $
goes to 0. Moreover, in a suitable class of convex planar domains, our estimate
turns out to be better than the one provided by the Payne-Weinberger
inequality.

Furthermore, we prove that, when
$p=n=2$ and
$\Omega $
consists of the points on one side of a smooth curve
$\gamma $, within a
suitable distance $\delta $
from it, then ${\mu}_{1}\left(\Omega \right)$
can be sharply estimated from below in terms of the length of
$\gamma $, the
${L}^{\infty}$ norm of its
curvature and $\delta $.

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