We prove that the curvature equation $\sum_{j=1}^n \alpha_j E_j(\tau_{\mathcal{M}})=\sum_{j=1}^n \alpha_j E_j(\tau_{\mathcal{N}})$ yields uniqueness up to translation for any two closed $C^2_+$ hypersurfaces $\mathcal{M}, \mathcal{N}\hookrightarrow\mathbb{R}^{n+1}$ whenever $(\alpha_1,\ldots,\alpha_n)\in\mathbb{R}_{\geq 0}^n\setminus\{0\}$ is log-concave and has no internal zeros. This gives an affirmative answer to a uniqueness question posed by Firey in the $C^2_+$ class.
We prove that for every even integer $N\geq 4$ or $N\in\{3,5,7\}$, every axially symmetric solution to the $Q$-curvature-type problem $$ \alpha P_N u + (N-1)!(1-\frac{e^{Nu}}{\int_{\mathbb{S}^N} e^{Nu}dw})=0 \ \ \ \ \ \mbox{on} \ \mathbb{S}^N $$ is constant, provided that $ \alpha\ge\frac{1}{2}$ and $\alpha \not =1$. T...
Chang-Feng Gui, Tuo Li, Jun-Cheng Wei et al.· 0 citations
Let $\Omega\subset\mathbb{R}^d$, $d\ge2$, be a bounded connected domain with boundary of class $C^{1,\alpha}$, where $0<\alpha<1$. For the adjoint Neumann--Poincar\'e operator $K^*_{\partial\Omega}$, normalised so that its distinguished eigenvalue is $1/2$, let $\lambda_j^+(\Omega)$ denote the upper min--max values on...
We study the Stokes operator with no-slip boundary conditions on the spaces $\mathrm{L}_{\sigma,n}(\Omega)$ and ${\mathrm{L}^1(\Omega,\mathbb{C}^d)}/{\nabla \mathrm{W}^{1,1}(\Omega,\mathbb{C})}$, where $\Omega\subset\mathbb{R}^d$ is an arbitrary bounded $\mathrm{C}^{1,\alpha}$-domain. We show that the Stokes operator o...
We prove the following inequality for Dirichlet polynomials: \[ \int_0^1 |f(1/2+\sigma)|\,d\sigma\lesssim \lim_{T\to\infty} \frac{1}{2T} \int_{-T}^T |f(it)| \, dt. \] In particular, for a Dirichlet series $f(s) = \sum_{n\geq 1} a_n n^{-s}$ belonging to the Hardy space $\mathscr{H}^1$ of Dirichlet series, \[ \left|a_1+\...
Let $\delta\in\mathbb{F}_{2^n}$ satisfy $\operatorname{Tr}_{\mathbb{F}_{2^n}/\mathbb{F}_2}(\delta)=1$. We study the permutation behavior of $$ f(x) = \left(\frac{1}{x^2+x+\delta}\right)^{2^k}+x $$ over $\mathbb{F}_{2^n}$. Helleseth and Zinoviev proved that $f(x)$ is a permutation for $k=0,1$, and remarked that numerica...
For the Dirichlet-type space $\mathcal D(\rho^{(1)},\rho^{(2)})$ on the unit bidisc $\mathbb D^2,$ where $\rho^{(1)}$ and $\rho^{(2)}$ are finite positive Borel measures on the closed unit disc $\overline{\mathbb D},$ we show that the L\'evy measure $\nu_{(\mathscr M_z,1)}$ associated with the completely alternating mu...
Santu Bera, S. S. Sequeira· 0 citations
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