Related papers: Tur\'an's extremal problem for positive definite f…

The constrained minimisers of convex integral functionals of the form $\mathscr F(v)=\int_\Omega F(\nabla^k v(x))\mathrm d x $ defined on Sobolev mappings $v\in \mathrm W^{k,1}_g(\Omega , \mathbb R^N )\cap K$, where $K$ is a closed convex…

Given a function $f$ on $\mathbb{F}_2^n$, we study the following problem. What is the largest affine subspace $\mathcal{U}$ such that when restricted to $\mathcal{U}$, all the non-trivial Fourier coefficients of $f$ are very small? For the…

We consider a family, $\mathcal{F}$, of subsets of an $n$-set such that the cardinality of the symmetric difference of any two elements $F,F'\in\mathcal{F}$ is not a multiple of 4. We prove that the maximal size of $\mathcal{F}$ is bounded…

The space $F^p$ ($1<p<\infty$) consists of all holomorphic functions $f$ on the open unit disk $\Bbb D$ such that $ \lim_{r\to 1}(1-r)^{1/q}\log^+M_{\infty}(r,f)=0,$ where $M_{\infty}(r,f)=\max_{\vert z\vert\le r}\vert f(z)\vert$ with…

Averaging linear functional on the space continuous functions of the group of diffeomorphisms of interval is found. Amenability of several discrete subgroups of the group of diffeomorphisms $\Diff^3([0,1])$ of interval is prove. In…

We consider shape optimization problems involving functionals depending on perimeter, torsional rigidity and Lebesgue measure. The scaling free cost functionals are of the form $P(\Omega)T^q(\Omega)|\Omega|^{-2q-1/2}$ and the class of…

We study the properties of the fundamental group of an affine curve over an algebraically closed field of characteristic $p$, from the point of view of embedding problems. In characteristic zero, the fundamental group is free, but in…

We look for solutions of $(-\Delta)^s u+f(u) = 0$ in a bounded smooth domain $\Omega$, $s\in(0,1)$, with a strong singularity at the boundary. In particular, we are interested in solutions which are $L^1(\Omega)$ and higher order with…

In this paper we obtain rigidity results for bounded positive solutions of the general capillary overdetermined problem \begin{equation} \left\{ \begin{array} {ll} \mathrm{div} \left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) + f(u) = 0…

We study the class ${\mathcal C}(\Omega)$ of univalent analytic functions $f$ in the unit disk $\mathbb{D} = \{z \in \mathbb{C} :\,|z|<1 \}$ of the form $f(z)=z+\sum_{n=2}^{\infty}a_n z^n$ satisfying \[ 1+\frac{zf"(z)}{f'(z)} \in \Omega,…

Consider the following inequalities due to Caffarelli, Kohn and Nirenberg {\it (Compositio Mathematica,1984):} $$\Big(\int_\Omega \frac{|u|^r}{|x|^s}dx\Big)^{\frac{1}{r}}\leq C(p,q,r,\mu,\sigma,s)\Big(\int_\Omega \frac{|\nabla…

For an analytic function $f$ defined on the unit disk $|z|<1$, let $\Delta(r,f)$ denote the area of the image of the subdisk $|z|<r$ under $f$, where $0<r\le 1$. In 1990, Yamashita conjectured that $\Delta(r,z/f)\le \pi r^2$ for convex…

We give bilateral pointwise estimates for positive solutions $u$ to the sublinear integral equation \[ u = \mathbf{G}(\sigma u^q) + f \quad \textrm{in} \,\, \Omega,\] for $0 < q < 1$, where $\sigma\ge 0$ is a measurable function, or a Radon…

This paper concentrates on positive definite functions on finite abelian groups, which are central to harmonic analysis and related fields. By leveraging the group structure and employing Fourier analysis, we establish a lower bound for the…

We consider autonomous integral functionals of the form $\mathcal F[u]:=\int_\Omega f(D u)\,dx$ with $u:\Omega\to\mathbb R^N$ $N\geq1$, where the convex integrand $f$ satisfies controlled $(p,q)$-growth conditions. We establish higher…

In this work, we obtain an existence of nontrivial solutions to a minimization problem involving a fractional Hardy-Sobolev type inequality in the case of inner singularity. Precisely, for $\lambda>0$ we analyze the attainability of the…

A topological group $G$ is extremely amenable if every continuous action of $G$ on a compact space has a fixed point. Using the concentration of measure techniques developed by Gromov and Milman, we prove that the group of automorphisms of…

It has been recently shown that for a convex domain $\Omega$ in $\mathbb C^n$ and $w\in\Omega$ the function $F_\Omega(w):=\big(K_\Omega(w)\lambda(I_\Omega(w))\big)^{1/n}$, where $K_\Omega$ is the Bergman kernel on the diagonal and…

Let $\big(\mathcal{S}_n^{\alpha,\kappa,\mathfrak{r}}(z)\big)_{n=1}^\infty$ be a sequence of the largest possible integer intervals, such that…

If a real harmonic function inside the open unit disk $B(0,1) \subset \mathbb{R}^2$ has its level set $\left\{x: u(x) = u(0)\right\}$ diffeomorphic to an interval, then we prove the sharp bound $\kappa \leq 8$ on the curvature of the level…