Let $\mathsf{d}\geq1$, $0<\mathsf{s}<\mathsf{d}$, and $N\geq1$. We prove that the optimal fixed-particle-number constant $\Lambda_N(\mathsf{s},\mathsf{d})$ in the Riesz Lieb--Oxford inequality is attained and that these constants are strictly increasing in $N$. The proof combines grand-canonical concentration--compactness with a strict one-particle extension. After recentering, a limiting plan arising from a maximizing sequence may assign positive probability to several particle numbers and hence be grand-canonical. A strict $N$-particle completion excludes this case, while the inequalities $\Lambda_N>\Lambda_k$ for $k<N$ exclude limits with a fixed lower particle number. Once attainment at particle number $N$ is known, the compact-support theorem of Di Marino and Lelotte arXiv:2607.11440, valid for all $0<\mathsf{s}<\mathsf{d}$, permits a non-product one-particle extension and yields $\Lambda_{N+1}(\mathsf{s},\mathsf{d})>\Lambda_N(\mathsf{s},\mathsf{d})$. Together, these implications close an induction beginning at $N=1$.
Let $\pi(\mathrm{d} x)\propto e^{-U(x)}\, \mathrm{d} x$ on $\mathbb{R}^d$, where $U$ is continuously differentiable and $m$-strongly convex with a globally $L$-Lipschitz gradient, $0<m\leq L<\infty$, and $\kappa=L/m$. Fixed-step Metropolis-adjusted Langevin algorithm (MALA) has known warm-start mixing-time upper bounds...
We study the central Hardy--Littlewood maximal operator $M_c$ in $\mathbb{R}^n$ for all $n\geq1$, with particular emphasis on the question of its injectivity. More precisely, we consider whether there exist two nonnegative functions $f,g\in L^1(\mathbb{R}^n)$ such that $\|f-g\|_1>0$ and $M_cf=M_cg$. Along the way, we o...
Given $\rho>0$, we consider the problem \[ \text{find $(\lambda,u) \in \mathbb{R} \times H_0^1(B)$ such that } \begin{cases} -\Delta u+\lambda u = |u|^{p-1}u&\text{in } B \\ \int_B u^2\,dx = \rho, \end{cases} \] where $B$ is a ball in $\mathbb{R}^N$, $N \ge 1$, and $1<p<2^*-1$. Without any further restriction on $N$, $...
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 $\omega_{\mathrm{FS}}$ be the normalized Fubini--Study form on $\mathbf P^n$, with $n\geq2$. We prove that the one-pole Lelong-number range in $DMA(\mathbf P^n,\omega_{\mathrm{FS}})$ is exactly $[0,1]$: for every $\lambda$ in this interval there is a Green function with a single pole, Monge--Amp\`ere measure equal...
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...
Qinyang Li· 0 citations
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