For $N\geq 1$, $p\geq 1$, and $\delta>0$, consider the nonlocal threshold functional \[ I_{\delta,p}(u)=\iint_{\{|u(x)-u(y)|>\delta\}} \frac{\delta^p}{|x-y|^{N+p}}\,\dd x\,\dd y. \] Nguyen and Squassina \cite{NguyenSquassina} asked whether $I_{\delta,p}$ decreases under Schwarz rearrangement. We give a negative answer in every dimension. For each $N\geq1$ and $\delta>0$, one and the same explicit counterexample works for every $p\geq1$: it is nonnegative, bounded, compactly supported, and takes only five values. Its four nontrivial superlevel sets are nested balls whose centers alternate between two points. Once the active threshold interactions are isolated, the energy difference reduces to comparing the interaction of a unit ball with a concentric annulus and with an eccentric shell. The sign is strict because the potential generated by the unit ball decreases with the radius. We also compute the energy gap in closed form in dimension one. This settles Open Problem~2.2 of Nguyen and Squassina.
Let $d,K,N\in \mathbb{N}$ with $K\geq 3$ and $d\geq 4K+4$. Let $\Delta\subset \mathbb{Z}^d$ be the vertex set of a nondegenerate $(K-1)$-simplex, and let $A\subseteq[N]^d$ contain no nontrivial similar copy of $\Delta$. We prove that \[ |A|\ll_{\Delta,d} N^d\exp\!\left(-c_{\Delta,d}\sqrt{\log N}\right) \] improving upo...
Andrew Lott, Á. Magyar, N. R. Ponagandla· 0 citations
We answer a question of Li--Liu--Xu: every $n$-dimensional klt germ $x\in(X,\Delta)$, where $n\ge2$, satisfies the sharp inequality \[ \widehat{\operatorname{vol}}(x,X,\Delta)\le n^{n-1}\operatorname{mld}_x(X,\Delta), \] with equality if and only if $\Delta=0$ near $x$, and analytically, $(x\in X)\cong\frac{1}{r}(1,\ld...
Let $\Omega_{n}=\{-1,1\}^n$ be the $n$-dimensional hypercube equipped with the normalized uniform measure, let $\nabla$ be the Walsh gradient and let $\Delta$ be the Walsh Laplacian. For every $1<p\leq 2$ we prove the following estimate \[ \|\nabla f\|_{L_p(\Omega_n;\ell_2^n)} \leq c_{{\rm abs}}(p-1)^{-2}\|\Delta^{1/p}...
Yong Jiao, Si-Jie Luo, D. Zanin et al.· 1 citation
We prove that for every dimension $n \geq 2$ and parameter $p \in ]1,n[$, the hedgehog $u_0(x) = x/|x|$ is the unique minimiser of the $p$-energy $\mathcal{E}_p [u]:=\int_{\mathbf{B}^n}|\nabla u|^p\,{\rm d}x$ in the class of $W^{1,p}$-mappings from the $n$-dimensional unit ball $\mathbf{B}^n$ to the $(n-1)$-dimensional...
Let $\Omega\subset\R^n$ be a domain and let $u\in W^{2,n}_{\loc}(\Omega)$ satisfy \[ \det D^2u\geq\delta>0\qquad\text{a.e. in }\Omega. \] Guerra and Tione asked whether the gradient mapping $Du$ must be open and discrete. We give a negative answer in every dimension $n\geq4$, in a form that isolates the sharp regularit...
Let $X$ be an isotropic log-concave random vector in $\mathbb{R}^n$ and let $G$ be standard Gaussian. Starting from the first moment comparison for gauges, we derive corresponding estimates at arbitrary moment orders. For every gauge $\phi$ and every $q\geqslant 1$, $$ \|\phi(G)\|_q\leqslant C\sqrt{\ln(en)+q}\,\|\phi(X...
