English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $\Omega \subset \mathbb{R}^2$ be a bounded convex domain in the plane andconsider \begin{align*}  -\Delta u &=1 \qquad \mbox{in}~\Omega \\ u &= 0 \qquad \mbox{on}~\partial\Omega. \end{align*} If $u$ assumes its maximum in $x_0 \in \Omega$, then theeccentricity of level sets close to the maximum is determined by the Hessian$D^2u(x_0)$. We prove that $D^2u(x_0)$ is negative definite and give aquantitative bound on the spectral gap $$ \lambda_{\max}\left(D^2u(x_0)\right)\leq - c_1\exp\left( -c_2\frac{diam(\Omega)}{inrad(\Omega)} \right)$$ foruniversal $c_1, c_2$ This is sharp up to constants. The proof is based on a newlower bound for Fourier coefficients whose proof has a topological component:if $f:\mathbb{T} \rightarrow \mathbb{R}$ is continuous and has $n$ signchanges, then $$ \sum_{k=0}^{n/2}{ \left| \left\langle f, \sin{kx}\right\rangle \right| + \left| \left\langle f, \cos{kx} \right\rangle \right| }\gtrsim_n \frac{ | f\|^{n+1}_{L^1(\mathbb{T})}}{ \| f\|^{n}_{L^{\infty}(\mathbb{T})}}.$$ This statement immediately implies estimates onhigher derivatives of harmonic functions $u$ in the unit ball: if $u$ is veryflat in the origin, then the boundary function $u(\cos{t}, \sin{t}):\mathbb{T}\rightarrow \mathbb{R}$ has to have either large amplitude or many roots. Italso implies that the solution of the heat equation starting with $f:\mathbb{T}\rightarrow \mathbb{R}$ cannot decay faster than $\sim\exp(-(\# \mbox{signchanges})^2 t/4)$.

Topological Bounds for Fourier Coefficients and Applications to Torsion